Mathematical Combinatorics (International Book Series), vol. 2 / 2018

ISBN 978-1-59973-575-7 978-1-59973-575-7 VOLUME 2,

MATHEMA MATHEMATICAL TICAL (INT (INTER ERNA NATI TION ONAL AL

2 01 8

COMBINATORICS COMBINATORICS BOOK BOOK

SERI SERIES ES))

Edited By Linfan MAO

THE MADIS OF CHINESE ACADEMY OF SCIENCES AND ACADEMY OF MATHEMATICAL COMBINATORICS & APPLICATIONS, USA Jun une e,

2018 2018

Vol.2, 2018

ISBN 978-1-59973-575-7

MATHEMA MATHEMATICAL TICAL (INT (INTER ERNA NATI TION ONAL AL

COMBINATORICS COMBINATORICS BOOK BOOK

SERI SERIES ES))

Edited By Linfan MAO

(www.mathcombin.com)

The Madis of Chinese Academy of Sciences and Academy of Mathematical Combinatorics & Applications, USA Jun une e,

2018 2018

Aims and Scope: The Mathematical The Mathematical Combinatorics (International Book Series) is a fully refereed international book series with ISBN number on each issue, sponsored by the MADIS of Chinese Academy of Sciences , sponsored by the MADIS of Chinese Academy of Sciences and and published in USA quarterly comprising 110-160 pages approx. per volume, which publishes publishes original original research research papers and survey articles articles in all aspects of Smarandache Smarandache multi-sp multi-spaces, aces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are: Smarandache multi-spaces with applications to other sciences, such as those of algebraic multi-systems, multi-metric spaces, , etc.. Smarandache geometries; Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and map enumeration; Combinatorial designs; Combinatorial enumeration; Different Differential ial Geometry; Geometry; Geometry Geometry on manifold manifolds; s; Low Dimension Dimensional al Topology; opology; Different Differential ial Topology; Topology of Manifolds; Geometrical aspects of Mathematical Physics and Relations with Manifold Topology; Applications of Smarandache multi-spaces to theoretical physics; Applications of Combinatorics to mathematics and theoretical physics; Mathematical theory on gravitational fields; Mathematical theory on parallel universes; Other applications of Smarandache multi-space and combinatorics. Generally, papers on mathematics with its applications not including in above topics are also welcome. It is also available from the below international databases:

···

Serials Group/Editorial Department of EBSCO Publishing 10 Estes St. Ipswich, MA 01938-2106, USA Tel.: (978) 356-6500, Ext. 2262 Fax: (978) 356-9371 http://www.ebsco.com/home/printsubs/priceproj.asp and

Gale Directory of Publications and Broadcast Media , Gale, a part of Cengage Learning 27500 Drake Rd. Farmington Hills, MI 48331-3535, USA Tel.: (248) 699-4253, ext. ext. 1326; 1-800-347-GALE 1-800-347-GALE Fax: Fax: (248) 699-8075 http://www.gale.com Indexing and Reviews: Mathematical Reviews: Mathematical Reviews (USA), Zentralblatt Math (Germany), Referativnyi Zhurnal (Russia), Mathematika (Russia), Directory of Open Access (DoAJ), International Statistical Institute (ISI), International Scientific Indexing (ISI, impact factor 1.743), Institute for Scientific Information (PA, USA), Library of Congress Subject Headings (USA). Subscription A Subscription A subscription can be ordered by an email directly to Prof.Linfan Mao,PhD The Editor-in-Chief of International Journal of Mathematical Combinatorics Chinese Academy of Mathematics and System Science Beijing, 100190, P.R.China Email: [email protected] [email protected]

Price: Price: US$48.00

Editorial Board (4th) Editor-in-Chief Linfan MAO Chinese Academy of Mathematics and System Science, P.R.China and Academy Academy of Mathemat Mathematical ical Combinat Combinatorics orics & Applications, USA Email: [email protected]

Shaofei Du Capital Normal University University, P.R.China Email: [email protected]

Deputy Editor-in-Chief

Yuanqiu Huang Hunan Normal University, P.R.China Email: [email protected]

Xiaodong Hu Chinese Academy of Mathematics and System Science, P.R.China P.R.China Email: [email protected]

Guohua Song Beijing Beijing Universit University y of Civil Engineering Engineering and H.Iseri Architecture, P.R.China Mansfield University, USA Email: [email protected] Email: hiseri@mnsfld.edu

Editors Arindam Bhattacharyya Jadavpur University, India Email: [email protected] Said Broumi Hassan II University Mohammedia Hay El Baraka Baraka Ben M’sik Casablanca Casablanca B.P.7951 Morocco

Xueliang Li Nankai University, P.R.China Email: [email protected] Guodong Liu Huizhou University Email: [email protected] W.B.Vasantha W.B.Vasantha Kandasamy Indian Institute of Technology, India Email: [email protected]

Junliang Cai Beijing Normal University University, P.R.China Email: [email protected]

Ion Patrascu Fratii Buzesti National College Craiova Romania

Yanxun Chang Beijing Jiaotong University University, P.R.China P.R.China Email: [email protected]

Han Ren East China Normal University, P.R.China Email: [email protected]

Jingan Cui Ovidiu-Ilie Sandru Beijing Beijing Universit University y of Civil Engineering Engineering and Politechnica University of Bucharest Architecture, P.R.China Romania Email: [email protected]

ii

Mathematical Mathematical Combinatorics Combinatorics (International (International Book Series)

Mingyao Xu Peking University, P.R.China Email: [email protected] Guiying Yan Chinese Academy of Mathematics and System Science, P.R.China Email: [email protected] Y. Zhang Department of Computer Science Georgia State University, Atlanta, USA

Famous Words: You can pay attention to the fact, in which case you’ll probably become a mathematician, or you can ignore it, in which case you’ll probably become a physicist. By Len Evans, a professor in Northwestern University.

Math.Combin.Bo Math.Combin.Book ok Ser. Vol.2(2018 Vol.2(2018), ), 1-12

Ricci Soliton and Conformal Ricci Soliton in Lorentzian β -Kenmotsu -Kenmotsu Manifold Tamalika Dutta (Department of Mathematics, Jadavpur University, Kolkata-700032, India)

Arindam Bhattacharyya (Department of Mathematics, Jadavpur University, Kolkata-700032, India) E-mail: [email protected], [email protected] [email protected]

Abstract: Abstract: In this paper we have studied quasi conformal curvature tensor, Ricci tensor, projective curvature tensor, pseudo projective curvature tensor in Lorentzian β -Kenmotsu -Kenmotsu manifold admitting Ricci soliton and conformal Ricci soliton.

Key Words: ords: Trans-Sasakian manifold, β -Kenmotsu -Kenmotsu manifold, Lorentzian Lorentzian β -Kenmotsu -Kenmotsu manifold, Ricci soliton, conformal Ricci flow.

AMS(2010): AMS(2010): 53C25, 35K65, 53C44, 53D10, 53D15. §1. Introduction

Hamilton started the study of Ricci flow [12] in 1982 and proved its existence. Hamilton existence. This concept concept was developed to answer Thurston’s geometric conjecture which says that each closed three manifold admits a geometric decomposition. Hamilton also [11]classified all compact manifolds with positive positive curvature curvature operator operator in dimension dimension four. Since then, then, the Ricci flow has become a powerful tool for the study of Riemannian manifolds, especially for those manifolds with positive curvature curvature.. Perelman Perelman also did an excellen excellentt work on Ricci flow [15], [16]. The Ricci flow equation is given by ∂g = ∂t

−2S

(1.1)

on a compact Riemannian manifold M with M with Riemannian metric g . A soluti solution on to the Ricc Riccii flow is called a Ricci soliton if it moves only by a one-parameter group of diffeomorphism and scaling.Ramesh Sharma [18], M. M. Tripathi [19], Bejan, Crasmareanu [4]studied Ricci soliton in contact metric manifolds also. The Ricci soliton equation is given by

£X g + 2S 2 S + + 2λg 2 λg = = 0,

(1.2)

where £X is the Lie derivative, S is S is Ricci tensor, g is Riemannian metric, X is X is a vector field and λ and λ is a scalar. The Ricci soliton is said to be shrinking, steady and expanding according as 1 The

first author is supported by DST ’Inspire’ ’Inspire’ of India. India. Reference Reference no: IF140748. IF140748. September 10, 2017, Accepted May 8, 2018.

2 Received

2

Tamalika Tamalika Dutta and Arindam Bhattacharyya Bhattacharyya

λ is negative, zero and positive respectively. In 2005, A.E. Fischer [10] introduced the concept of conformal Ricci flow flow which is a variation of the classical Ricci flow equation. In classical Ricci flow equation the unit volume constraint plays an important role but in conformal Ricci flow equation scalar curvature R is considered as constraint. constraint. Since the conformal conformal geometry plays plays an important important role to constrain constrain the scalar curvature and the equations are the vector field sum of a conformal flow equation and a Ricci flow equation, equation, the resulting resulting equations equations are named as the conformal conformal Ricci flow equations. equations. The conformal conformal Ricci flow equation equation on M M where M M is considered as a smooth closed connected oriented n-manifold(n n-manifold( n > 3), is defined by the equation [10] ∂g g + 2(S 2(S + + ) = pg ∂t n

−

(1.3)

and r and r = = 1, where p where p is is a scalar non-dynamical field(time dependent scalar field), r is the scalar curvature of the manifold and n is the dimension of manifold.

−

In 2015, N. Basu and A. Bhattacharyya [3] introduced the notion of conformal Ricci soliton and the equation is as follows

£X g + 2S 2 S = = [2λ [2λ

2 − ( p + p + )]g. )]g. n

(1.4)

The equation is the generalization of the Ricci soliton equation and it also satisfies the conformal Ricci flow equation. An almost contact metric structure (φ,ξ,η,g ( φ,ξ,η,g)) on a manifold M is M is called a trans-Sasakian structure structure [14] if the product manifold manifold belongs to the class W 4 where W where W 4 is a class of Hermitian manifolds which are closely related to locally conformal Kaehler manifolds [6]. A trans-Sasakian structure of type (0, (0 , 0), 0), (0, (0, β ) and (α, (α, 0) are cosymplectic [5], β Kenmotsu [13], and α and α Sasakian [13], respectively. respectively.

−

−

§2. Preliminaries

A differentiable manifold of dimension n is n is called Lorentzian Kenmotsu manifold [2] if it admits a (1, (1, 1) tensor field φ, a covarien covarientt vector vector field ξ , a 1-form η and Lorentzian metric g which satisfy on M on M respectively respectively such that φ2 X = = X + + η (X )ξ, g (X, ξ ) = η( η (X ), η (ξ ) =

−1, η(φX ) = 0, φξ = = 0,

(2.1)

g (φX,φY ) φX,φY ) = g( g (X, Y ) Y ) + η(X )η (Y ) Y ),

(2.2)

(2.3)

for all X all X,, Y χ( χ(M ) M ).

∈ ∈

If Lorentzian Kenmotsu manifold M satisfies M satisfies

∇X ξ = β − η(X )ξ ], (∇X φ)Y = β (g(φX,Y ) = β [X − φX,Y )ξ − η(Y ) Y )φX ),

(2.4)

3

Ricci Soliton and Conformal Ricci Soliton in Lorentzian β -Kenmotsu Manifold

∇X η)Y = αg( αg (φX,Y ) φX,Y ),

(

∇

(2.5)

where denotes the operator of covariant differentiation with respect to the Lorentzian metric g. Then the manifold M is M is called Lorentzian β Kenmotsu manifold.

−

Furthermore, on an Lorentzian β -Kenmotsu -Kenmotsu manifold M the M the following relations hold [1], [17]: η(R(X, Y ) Y )Z ) = β = β 2 [g (X, Z )η (Y ) Y )

− g(Y, Z )η(X )], )],

R(ξ, X )Y = β 2 [η (Y )X )X

− − g(X, Y ) Y )ξ ], − η(Y ) R(X, Y ) Y )ξ = β = β 2 [η (X )Y − Y )X ], S (X, ξ ) = −(n − 1)β 1)β 2 η (X ), Qξ = = −(n − 1)β 1)β 2 ξ, S (ξ, ξ ) = (n − 1)β 1)β 2 ,

(2.6) (2.7) (2.8) (2.9)

(2.10)

(2.11)

where β where β is some constant, R is the Riemannian curvature tensor, S is S is the Ricci tensor and Q is the Ricci operator given by S ( S (X, Y ) Y ) = g( g (QX,Y ) QX,Y ) for all X, X , Y χ( χ(M ). ).

∈ ∈

Now from definition of Lie derivative we have (£ξ g )(X, )(X, Y ) Y ) = (

∇ξ g)(X, )(X, Y ) Y ) + g (β [X − − η(X )ξ ], Y ) Y ) + g (X, β [Y − − η(Y ) Y )ξ ]) ]) 2βg( βg (X, Y ) Y ) − 2βη( βη (X )η (Y ) Y ). (2. (2.12)

=

Applying Ricci soliton equation (1 .2) in (2. (2.12) we get 1 1 [ 2λg( [2βg [2βg((X, Y ) λg(X, Y )] Y )] Y ) 2βη( βη (X )η (Y )] Y )] 2 2 = λg( λg (X, Y ) Y ) βg( βg (X, Y ) Y ) + βη( βη (X )η (Y ) Y ) ´ (X, Y ) = Ag( Ag Y ) + βη( βη (X )η (Y ) Y ),

S (X, Y ) Y ) =

−

−

−

−

−

(2. (2.13)

where A´ = ( λ

− − β ), which shows that the manifold is η-Einstein. η -Einstein.

Also ´ + QX = AX + βη( βη (X )ξ, ´ + β )η (X ) = Aη S (X, ξ ) = (A Aη((X ).

(2.14)

(2.15)

{ }

If we put X = Y = e i in (2. (2 .13) where ei is the orthonormal basis of the tangent space T M where M where T T M is M is a tangent bundle of M of M and and summing over i, i , we get ´ + β. R(g ) = An

Proposition 2 Proposition 2..1 A Lorentzian β -Kenmotsu manifold admitting Ricci soliton is η-Einstein.

4


Again applying conformal Ricci soliton (1. (1 .4) in (2. (2 .12) we get 1 2 1 [2λ [2λ ( p + )]g )]g (X, Y ) Y ) [2βg [2βg((X, Y ) Y ) 2 n 2 ´ (X, Y ) = Bg( Bg Y ) + βη( βη (X )η (Y ) Y ),

S (X, Y ) Y ) =

−

−

− 2βη( βη (X )η (Y )] Y )] (2. (2.16)

where

2 ´ = 1 [2λ B [2λ ( p + p + )] β, 2 n which also shows that the manifold is η-Einstein. η -Einstein. Also ´ + QX = = BX + βη( βη (X )ξ,

−

−

(2.17)

(2.18)

´ + β )η (X ) = Bη S (X, ξ ) = ( B B η (X ).

(2.19)

If we put X = Y = e i in (2. (2 .16) where ei is the orthonormal basis of the tangent space T M where M where T T M is M is a tangent bundle of M of M and and summing over i, i , we get

{ }

´ + β. r = Bn

For conformal Ricci soliton r( r (g ) =

−1. So ´ + β −1 = Bn + Bn

which gives B gives B = n1 ( β 1). 1). Comparing the values of B of B from (2. (2 .17) with the above equation we get

− −

λ =

1 (β (n n

− 1) − 1) + 12 ( p + n2 )

Lorentzian β -Kenmots -Kenmotsu u manifold manifold admittin admittingg confor conformal mal Ricci Ricci soliton soliton is Proposition 2.2 A Lorentzian η -Einstein and the value of the scalar λ =

1 (β (n n

2 − 1) − 1) + 12 ( p + p + ). n

-Kenmotsu Manifold Admitting Ricci §3. Lorentzian β -Kenmotsu ˜ = Soliton, Conformal Ricci Soliton and R(ξ, X ).C = 0 Let M be a n dimensional Lorentzian β -Kenmotsu β -Kenmotsu manifold admitting Ricci soliton ( g , V , λ). λ). ˜ on M Quasi conformal curvature tensor C on M is is defined by ˜ (X, Y ) C Y )Z = aR( aR(X, Y ) Y )Z + + b[S (Y, Z )X S (X, Z )Y + g (Y, Z )QX r a [ ][ + 2b 2 b][g ][g (Y, Z )X g (X, Z )Y ] Y ], 2n + 1 2n

− − g(X, Z )QY QY ]]

− −

−

where r where r is scalar curvature.

− −

(3. (3.1)

5


Putting Z Putting Z = = ξ in ξ in (3. (3.1) we have ˜ (X, Y ) C Y )ξ = aR( aR(X, Y ) Y )ξ + + b[S (Y, ξ )X S (X, ξ )Y + g (Y, ξ )QX r a [ ][ + 2b 2 b][g ][g (Y, ξ )X g (X, ξ )Y ] Y ]. 2n + 1 2n

− −

−

− − g(X, ξ )QY QY ]]

− −

(3. (3.2)

Using (2. (2.1), 1), (2. (2.8), 8), (2. (2.14), 14), (2. (2.15) in (3. (3 .2) we get ˜ (X, Y ) ´ C Y )ξ = = [ aβ 2 + Ab + Ab + Ab

−

Let D =

− [ 2n r+ 1 ][ 2an + 2b 2 b]](η ]](η (Y ) Y )X − − η(X )Y ) Y ).

´ − [ r ][ a + 2b −aβ 2 + Ab + Ab 2 b], 2 n + 1 2n

so we have ˜ (X, Y ) C Y )ξ = D = D((η (Y )X )X

− − η(X )Y ) Y ).

(3.3)

Taking inner product with Z in (3.3) we get Z in (3. ˜ (X, Y ) −η(C Y )Z ) = D = D[[η (Y ) Y )g (X, Z ) − η (X )g (Y, Z )]. )].

(3.4)

Now we consider that the Lorentzian β -Kenmotsu β -Kenmotsu manifold M which M which admits Ricci soliton ˜ is quasi conformally semi symmetric i.e. R(ξ, X ).C = = 0 holds in M , M , which implies ˜ (Y, Z )W ) R(ξ, X )( )(C W )

− C ˜ (R(ξ, X )Y, Z )W − − C ˜ (Y, R(ξ, X )Z )W − − C ˜ (Y, Z )R(ξ, X )W = 0,0 , (3.5)

for all vector fields X,Y,Z,W on M on M ..

Using (2. (2.7) in (3. (3.5) and putting W = ξ we ξ we get ˜ (Y, Z )ξ )X g (X, C ˜ (Y, Z )ξ )ξ η (Y ) ˜ (X, Z )ξ + ˜ (ξ, Z )ξ η (C Y )C + g (X, Y ) Y )C ˜ (Y, X )ξ + ˜ (Y, ξ )ξ η (ξ )C ˜ (Y, Z )X + ˜ (Y, Z )ξ = η (Z )C + g (X, Z )C + g (X, ξ )C = 0.

− −

− −

−

(3. (3.6)

Taking inner product with ξ in ξ in (3. (3 .6) and using (2. (2 .2), 2), (3. (3.3) we obtain ˜ (Y, Z )ξ ) + η (C ˜ (Y, Z )X ) = 0 . g (X, C

(3.7)

Putting Z Putting Z = = ξ in ξ in (3. (3.7) and using (3. (3 .3) we get ˜ (Y, Z )X ) = 0. −Dg( Dg (X, Y ) Y ) − Dη( Dη (X )η (Y ) Y ) + η (C

(3.8)

Now from (3. (3.1) we can write ˜ (Y, ξ )X = aR( C aR(Y, ξ )X + + b[S (ξ, X )Y S (Y, X )ξ + + g (ξ, X )QY r a [ ][ + 2b 2 b][g ][g (ξ, X )Y g (Y, X )ξ ]. 2n + 1 2n

− − g(Y, X )Qξ ]

− −

−

− −

(3. (3.9)

6


Taking inner product with ξ and ξ and using (2. (2 .2), 2), (2. (2.7), 7), (2. (2.9), 9), (2. (2.10) in (3. (3.9) we get ˜ (Y, ξ )X ) = aη( η (C aη (β 2 (g (X, Y ) Y )ξ

´ (Y ) − η(X )Y )) Y )) + b[Aη( Aη(X )η (Y ) Y ) + S (X, Y ) Y ) + η(X )( )(Aη( Aη Y ) r a ´ + β )] −βη( βη (Y )) Y )) − g (X, Y )( Y )(−A )] − [ ][ + 2b][η ][η (X )η (Y ) Y ) + g (X, Y )] Y )].. 2n + 1 2n

After a long simplification we have

− bβ − − aβ 2 − [ 2n r+ 1 ][ 2an + 2b 2 b]] ´ − aβ 2 − [ r ][ a + 2b +η (X )η (Y )[2 Y )[2Ab 2 b]] + bS (X, Y ) Y ). 2 n + 1 2n

˜ (Y, ξ )X ) = g (X, Y )[ ´ η (C Y )[Ab

(3. (3.10)

Putting (3. (3.10) in (3. (3.5) we get ) = S (X, Y ) ρg( ρg (X, Y ) Y ) + ση( ση (X )η (Y ) Y ), where ρ =

1 [D + bβ + + aβ 2 b

and σ =

1 [D + aβ 2 b

(3.11)

´ + [ r ][ a + 2b − Ab 2 b]] 2n + 1 2n

´ + [ r ][ a + 2b − 2Ab 2 b]]. ]]. 2n + 1 2n

So from (3. (3 .11) we conclude that the manifold becomes η -Einstein -Einstein manifold. manifold. Thus we can write the following theorem:

-Kenmotsu manifold admits Ricci soliton and is quasi conforTheorem 3.1 If a Lorentzian β -Kenmotsu ˜ = 0, then the manifold is η -Einstein manifold where C ˜ mally semi symmetric i.e. R(ξ, X ).C is quasi conformal curvature tensor and R(ξ, X ) is derivation of tensor algebra of the tangent space of the manifold. If a Lorentzian β -Kenmotsu -Kenmotsu manifold admits conformal Ricci soliton then after a brief calculati calculation on we can also establish establish that the manifold manifold becomes η-Einstein, -Einstein, only the values values of constants ρ, constants ρ, σ will be changed which would not hamper our main result. Hence we can state the following theorem:

-Kenmotsu manifold admitting conformal Ricci soliton and is Theorem 3.2 A Lorentzian β -Kenmotsu ˜ = quasi conformally semi symmetric i.e. R(ξ, X ).C = 0, then the manifold is η -Einstein manifold ˜ is where C is quasi conformal curvature tensor and R( R (ξ, X ) is derivation of tensor algebra of the tangent space of the manifold.

-Kenmotsu Manifold Admitting Ricci §4. Lorentzian β -Kenmotsu Soliton, Conformal Ricci Soliton and R(ξ, X ).S = = 0 Let M be a n dimensional Lorentzian β -Kenmotsu β -Kenmotsu manifold admitting Ricci soliton ( g , V , λ). λ). Now we consider that the tensor derivative of S of S by R by R((ξ, X ) is zero i.e. R(ξ, X ).S = = 0. Then the

7


Lorentzian β Lorentzian β -Kenmotsu -Kenmotsu manifold admitting Ricci soliton is Ricci semi symmetric which implies S (R(ξ, X )Y, Z ) + S (Y, R(ξ, X )Z ) = 0 .

(4.1)

Using (2. (2.13) in (4. (4.1) we get ´ (R(ξ, X )Y, Z ) + βη( ´ (Y, R(ξ, X )Z ) + βη( Ag( Ag βη (R(ξ, X )Y ) Y )η (Z ) + Ag( Ag βη (Y )η )η (R(ξ, X )Z ) = 0. (4.2) Using (2. (2.7) in (4. (4.2) we get ´ (β 2 [η (Y ) Ag( Ag Y )X

´ (Y, β 2 [η (Z )X − − − g(X, Y ) − g(X, Z )ξ ])]) + βη( Y )ξ ], Z ) + Ag( Ag βη (β 2 [η (Y ) Y )X −

g (X, Y ) Y )ξ ])η ])η (Z ) + βη( βη (Y ) Y )η (β 2 [η (Z )X

− − g(X, Z )ξ ])]) = 0.0.

(4.3)

Using (2. (2.2) in (4. (4.3) we have ´ 2 η (Y ) Aβ Y )g (X, Z )

´ 2 η (Z )g (X, Y ) ´ 2 η (Z )g (X, Y ) ´ 2 η (Y ) − Aβ Y ) + Aβ Y ) − Aβ Y )g (X, Z )

+β 3 η (Y ) Y )η (X )η (Z ) + β 3 g (X, Y ) Y )η (Z ) + β 3 η (Y ) Y )η (X )η (Z ) + β 3 g (X, Z )η (Y ) Y ) = 0. 0.

(4. (4.4)

Putting Z Putting Z = = ξ in ξ in (4. (4.4) and using (2. (2 .2) we obtain g (X, Y ) Y ) =

−η(X )η(Y ) Y ).

Hence we can state the following theorem:

-Kenmotsu manifold admits Ricci soliton and is Ricci semi Theorem 4.1 If a Lorentzian β -Kenmotsu symmetric i.e. R(ξ, X ).S = = 0, then g( g (X, Y ) Y ) = η (X )η (Y ) Y ) where S S is Ricci tensor and R( R (ξ, X ) is derivation of tensor algebra of the tangent space of the manifold.

−

If a Lorentzian β -Kenmotsu β -Kenmotsu manifold admits conformal Ricci soliton then by similar calculation culation we can obtain the same result. result. Hence Hence we can state the following following theorem:

-Kenmotsu manifold admitting conformal Ricci soliton and is Theorem 4.2 A Lorentzian β -Kenmotsu Ricci semi symmetric i.e. R(ξ, X ).S = = 0, then g (X, Y ) Y ) = η (X )η (Y ) Y ) where S is Ricci tensor and R( R (ξ, X ) is derivation of tensor algebra of the tangent space of the manifold.

−

-Kenmotsu Manifold Admitting Ricci §5. Lorentzian β -Kenmotsu Soliton, Conformal Ricci Soliton and R(ξ, X ).P = 0 Let M be a n dimensional Lorentzian β -Kenmotsu β -Kenmotsu manifold admitting Ricci soliton ( g , V , λ). λ). The projective curvature tensor P on M on M is is defined by

− − 21n [S (Y, Z )X − − S (X, Z )Y ] Y ].

P ( P (X, Y ) Y )Z = R( R (X, Y ) Y )Z

(5.1)

Here we consider that the manifold is projectively semi symmetric i.e. R(ξ, X ).P = = 0 holds.

8


So R(ξ, X )(P )(P ((Y, Z )W ) W )

− P ( P (R(ξ, X )Y, Z )W − − P ( P (Y, R(ξ, X )Z )W − − P ( P (Y, Z )R(ξ, X )W = 0, 0 , (5.2)

for all vector fields X,Y,Z,W on M on M .. Using (2. (2.7) and putting Z putting Z = = ξ in ξ in (5. (5.2) we have

− − g(X, P ( P (Y, ξ )W ) W )ξ − η (Y ) Y )P ( P (X, ξ )W + g (X, Y ) Y )P ( P (ξ, ξ )W −η(ξ )P ( − η(W ) P (Y, X )W + g (X, ξ )P ( P (Y, ξ )W − W )P ( P (Y, ξ )X + + g (X, W ) W )P ( P (Y, ξ )ξ = = 0. η (P ( P (Y, ξ )W ) W )X

(5. (5.3)

Now from (5. (5.1) we can write

− − n −1 1 [S (ξ, Z )X − − S (X, Z )ξ ].

P ( P (X, ξ )Z = = R( R (X, ξ )Z

(5.4)

Using (2. (2.7), 7), (2. (2.15) in (5. (5.4) we get P ( P (X, ξ )Z = = β 2 g (X, Z )ξ + +

1 n

−1

S (X, Z )ξ + + (

A n

2

− 1 − β )η(Z )X.

(5.5)

Putting (5. (5.5) and W and W = ξ in ξ in (5. (5.3) and after a long calculation we get 1 n

−1

S (X, Y ) Y )ξ + + (

A n

−1

− − n A− 1 g(X, Y ) Y )ξ

+ β 2 )η (X )Y

−( n A− 1 + β 2)η(Y ) Y )X = 0. 0.

(5.6)

Taking inner product with ξ in ξ in (5. (5 .6) we obtain S (X, Y ) Y ) =

−Ag( Ag(X, Y ) Y ),

which clearly shows that the manifold in an Einstein manifold. Thus we can conclude the following theorem:

-Kenmotsu manifold admits Ricci soliton and is projectively Theorem 5.1 If a Lorentzian β -Kenmotsu semi symmetric i.e. R(ξ, X ).P = 0 holds, then the manifold is an Einstein manifold where P is projective curvature tensor and R(ξ, X ) is derivation of tensor algebra of the tangent space of the manifold. If a Lorentzian β Lorentzian β -Kenmotsu -Kenmotsu manifold admits conformal Ricci soliton then using the same calculation we can obtain similar result, only the value of constant A will be changed which would would not hamper our main result. result. Hence we can state the following following theorem: theorem:

-Kenmotsu manifold admitting conformal Ricci soliton and is Theorem 5.2 A Lorentzian β -Kenmotsu projecti projectively vely semi symmetric i.e. R(ξ, X ).P = 0 holds, then the manifold is an Einstein manifold where P P is projective curvature tensor and R( R (ξ, X ) is derivation of tensor algebra of the tangent space of the manifold.

9


§6. Lorentzian β -Kenmotsu -Kenmotsu Manifold Admitting Ricci

˜ =0 Soliton, Conformal Ricci Soliton and R(ξ, X ).P

Let M be a n dimensional Lorentzian β -Kenmotsu β -Kenmotsu manifold admitting Ricci soliton ( g , V , λ). λ). ˜ The pseudo projective curvature tensor P on M on M is is defined by ˜ (X, Y ) P ( P Y )Z = aR( aR (X, Y ) Y )Z + + b[S (Y, Z )X

− − S (X, Z )Y ] Y ]

− nr [ n −a 1 + b][g − g(X, Z )Y ] ][g (Y, Z )X − Y ].

(6.1)

˜=0 Here we consider that the manifold is pseudo projectively semi symmetric i.e. R(ξ, X ).P holds. So ˜ (Y, Z )W ) R(ξ, X )( )(P ( P W )

˜ (R(ξ, X )Y, Z )W − ˜ (Y, R(ξ, X )Z )W − ˜ (Y, Z )R(ξ, X )W = 0, − P ( − P ( − P ( P P P 0 , (6.2)

for all vector fields X,Y,Z,W on M on M .. Using (2. (2.7) and putting W putting W = ξ in ξ in (6. (6 .2) we have ˜ (Y, Z )ξ )X η (P ( P

˜ (Y, Z )ξ )ξ − η (Y ) ˜ (X, Z )ξ + ˜ (ξ, Z )ξ − − g(X, P ( P Y )P ( P + g (X, Y ) Y )P ( P

˜ (Y, X )ξ + ˜ ˜ (Y, Z )X + ˜ (Y, Z )ξ = −η(Z )P ( + g (X, Z )P ( + η (X )P ( = 0. P P (Y, ξ )ξ − η (ξ )P ( P P

(6.3)

Now from (6. (6.1) we can write r a − − S (X, ξ )Y ] Y ] + [ + b][g ][g (Y, ξ )X − − g(X, ξ )Y ] Y ]. (6.4) n n−1

˜ (X, Y ) P ( P Y )ξ = aR = aR((X, Y ) Y )ξ + + b[S (Y, ξ )X

Using (2. (2.1), 1), (2. (2.8), 8), (2. (2.15) in (6. (6 .4) and after a long calculation we get ˜ (X, Y ) P ( P Y )ξ = ϕ = ϕ((η (X )Y

− − θ(Y ) Y )X ),

where ϕ where ϕ = = (aβ 2

(6.5)

− Ab − nr [ n−a 1 + b]). ]).

Using (6. (6.5) and putting Z putting Z = = ξ in ξ in (6. (6.3) we obtain ˜ (Y, ξ )X + P ( P + ϕη( ϕη(X )Y

− − ϕg( ϕg (X, Y ) Y )ξ = = 0.

(6.6)

Taking inner product with ξ in ξ in (6. (6 .6) we get ˜ (Y, ξ )X ) + ϕη( η (P ( P ϕη(X )η (Y ) Y )

− ϕg( ϕg (X, Y ) Y ) = 0.

(6.7)

Again from (6. (6 .1) we can write

− − S (X, Z )ξ ] + nr [ n −a 1 + b][g ][g (ξ, Z )X − − g(X, Z )ξ ]. (6.8)

˜ (X, ξ )Z = P ( P = a( a (X, ξ )Z + + b[S (ξ, Z )X

10


Using (2. (2.1), 1), (2. (2.7), 7), (2. (2.15) in (6. (6 .8) we get ˜ (X, ξ )Z = P ( P = aβ 2 [g (X, Z )ξ

− η(Z )X ] + b[Aη( Aη(Z )X − − S (X, Z )ξ ]

r a + [ + b][g ][g (ξ, Z )X n n 1

− − g(X, Z )ξ ].

−

(6.9)

Taking inner product with ξ and ξ and replacing X replacing X by Y by Y ,, Z by X by X in in (6. (6.9) we have ˜ (Y, ξ )X ) = aβ 2 [ g (X, Y ) η (P ( P Y )

−

− η(X )η(Y )] Y )] + b[Aη( Aη(X )η(Y ) Y ) + S (X, Y )]+ Y )]+

r a [ + b][η ][η (X )η (Y ) Y ) n n 1

−

− g(X, Y )] Y )]..

(6.10)

(6.11)

Using (6. (6.10) in (6. (6.7) and after a brief simplification we obtain S (X, Y ) Y ) = T = T g (X, Y ) Y ) + U η (X )η (Y ) Y ),

a 2 − 1b [−aβ 2 − nr [ n−a 1 + b] − ϕ] and U and U = − 1b [ϕ + nr [ n− 1 + b] + Ab − aβ ].

where T where T =

From (6. (6.11) we can conclude that the manifold is η -Einstein. Thus we have the following following theorem:

-Kenmotsu manifold admits Ricci soliton and is pseudo proTheorem 6.1 If a Lorentzian β -Kenmotsu ˜ = 0 holds, then the manifold is η Einstein manifold jectively jectively semi symmetric i.e. R(ξ, X ).P ˜ is pseudo projective curvature tensor and R( where P R (ξ, X ) is derivation of tensor algebra of the tangent space of the manifold. If a Lorentzian β Lorentzian β -Kenmotsu -Kenmotsu manifold admits conformal Ricci soliton then by following the same calculation we would obtain the same result, only the constant value of T of T and U and U will will be changed. changed. Hence we can state the following following theorem:

-Kenmotsu manifold admitting conformal Ricci soliton and is Theorem 6.2 A Lorentzian β -Kenmotsu ˜ = 0 holds, then the manifold is η Einstein pseudo projectively semi symmetric i.e. R(ξ, X ).P ˜ is pseudo projective curvature tensor and R(ξ, X ) is derivation of tensor manifold where P algebra of the tangent space of the manifold.

§7. An Example of a 3-Dimensional Lorentzian β -Kenmotsu -Kenmotsu Manifold

In this section we construct an example of a 3-dimensional Lorentzian β -kenmotsu -kenmotsu manifold.To construct this, we consider the three dimensional manifold M = (x,y,z) x,y,z) R 3 : z = 0 where (x,y,z) x,y,z ) are the standard coordinates in R 3 . The vector fields

{

= e −z e1 = e

∂ ∂ ∂ = e −z , e3 = e = e −z , e2 = e ∂x ∂y ∂z

are linearly independen independentt at each each point of M of M..

∈

 }

11


Let g Let g be the Lorentzian metric defined by g (e1 , e1 ) = 1, g (e2 , e2) = 1, 1 , g (e3, e3 ) =

− 1,

g (e1 , e2 ) = g( g (e2 , e3 ) = g = g((e3 , e1 ) = 0. Let η Let η be the 1-form which satisfies the relation η(e3 ) =

− 1.

Let φ be the (1, (1 , 1) tensor field defined by φ(e1 ) = have

−e2, φ(e2) = −e1, φ(e3) = 0. Then we

φ2 (Z ) = Z + + η (Z )e3 , g (φZ,φW ) = g( g (Z, W ) W ) + η (Z )η (W ) W ) for any Z, W χ( χ (M 3 ). Thus for e3 = ξ, ξ , (φ,ξ,η,g) φ,ξ,η,g) defines an almost contact metric structure on M on M.. Now, after calculating we have

∈ ∈

[e1 , e3 ] = e −z e1 , [e1 , e2 ] = 0, [e2 , e3 ] = e −z e2 . The Riemannian connection

∇ of the metric is given by the Koszul’s formula which is

2g (

∇X Y, Z ) = X g(Y, Z ) + Y g(Z, X ) − Zg (X, Y ) Y ) −g(X, [Y, Z ])]) − g(Y, [X, Z ])]) + g(Z, [X, Y ]) Y ])..

(7.1)

By Koszul’s formula we get

∇e e1 = e = e−z e3 , ∇e e1 = 0, ∇e e1 = 0, 1

2

3

∇e e2 = 0, ∇e e2 = ‘e−z e3, ∇e e2 = 0, ∇e e3 = e−z e1, ∇e e3 = e = e−z e2 , ∇e e3 = 0. 1

1

2

3

2

3

From the above we have found that β that β = = e −z and it can be easily shown that M 3 (φ,ξ,η,g) φ,ξ,η,g) is a Lorentzian β -kenmots -kenmotsu u manifold. manifold. The results establish established ed in this note can be verified verified on this manifold.

References [1] C.S. C.S. Bag Bagew ewadi adi and E. Girish Girish Kumar, Kumar, Note Note on transtrans-Sas Sasaki akian an manif manifold olds, s, Tensor.N.S., (65)(1) (2004), 80-88. [2] N.S. Basavarajappa, Basavarajappa, C.S. Bagewadi, Bagewadi, D.G. Prakasha, Prakasha, Some results on :orentzia :orentzian n β - Ken Annals of the Universit Universityy of Craiov Craiova, a, Math. Comp. Comp. Sci. Ser. , (35) 2008, motsu manifolds, Annals 7-14.

12


[3] Nirabhra Nirabhra Basu, Arindam Bhattacharyy Bhattacharyya, a, Conformal Ricci soliton in Kenmotsu Kenmotsu manifold, manifold, Global Journal of Advanced Research on Classical and Modern Geometries , (2015), 15-21. [4] Bejan, Crasmarean Crasmareanu, u, Ricci Ricci solitons solitons in manifold manifoldss with quasi-const quasi-constant ant curvature, curvature, Publ. Math. Debrecen Debrecen , (78)(1) (2011), 235-243. [5] D. E. Blair, Contact Contact manifolds manifolds in Riemannian Riemannian geometry geometry,, Lectures notes in Mathematics , Springer-Verlag, Berlin, (509) (1976), 146. [6] S. Dragomir and L. Ornea, Locallly conformal Kaehler geometry geometry,, Progress in Mathematics , 155, Birkhauser Birkhauser Boston, Inc., Boston, Boston, MA, 1998. [7] T. Dutta, N. Basu, A. Bhattach Bhattacharyya aryya,, Some curvature curvature identiti identities es on an almo almost st conformal gradient shrinking RICCI soliton, Journal of Dynamical Systems and Geometric Theories , (13)(2) (2015), 163-178. [8] Tamalika amalika Dutta, Arindam Bhattacharyya and Srabani Debnath, Conformal Ricci Soliton International J.Math. Combin., (3) (2016), 17-26. in Almost C(λ C( λ) Manifold, International [9] T. Dutta, N. Basu, A. Bhattacharyy Bhattacharyya, a, Almost conformal conformal Ricci solitons on 3-dimensio 3-dimensional nal trans-Sasakian manifold, Hacettepe Journal of Mathematics and Statistics , (45)(5) (2016), 1379-1392. [10] A. E. Fischer, An introduction introduction to conformal Ricci Ricci flow, Class.Quantum Grav. , (21) (2004), S171-S218. [11] R. S. Hamilton, Hamilton, The Ricci flow on surfaces, surfaces, Contemporary Mathematics , (71) (1988), 237261. [12] R. S. Hamilton, Three manifold manifold with positive Ricci curvature, curvature, J.Differential Geom., (17)(2) (1982), 255-306. Tohoku ku Math. J., (24) [13] K. Kenmotsu, Kenmotsu, A class of almost almost contact contact Riemannian Riemannian manifolds, manifolds, Toho (1972), 93-103. [14] A. J. Oubina, Oubina, New classes classes of almost almost Contact metric metric structures, structures, Publ.Math.Debrecen , (32) (1985),187-193. [15]] G. Perel [15 Perelman man,, The entro entropy py formu formula la for the Ricci Ricci flow flow and its its geo geomet metric ric applic applicati ations ons,, http://arXiv.org/abs/math/0211159 , (2002) 1-39. [16] G. Perelman,Ric Perelman,Ricci ci flow with surgery surgery on three manifolds, manifolds, http://arXiv.org/abs/math/ 0303109 , (2003) 1-22. [17] D.G. Prakasha, C.S. Bagewadi and N.S. Basavarajappa, On Lorentzian β Lorentzian β -Kenmotsu -Kenmotsu manifolds, Int.Journal of Math. , (19)(2) (19)(2) (2008), (2008), 919-927. [18] Ramesh Sharma, Almost Ricci solitons and K-contact geometry, geometry, Monatsh Math. , (2014), 175:621-628. [19] M. M. Tripathi, Tripathi, Ricci solitons in contact metric manifolds, arXiv:0801,4222v1 , [mathDG], (2008).


Some Properties of Conformal β -Change H.S.Shukla and Neelam Mishra (Department of Mathematics and Statistics, D. D. U. Gorakhpur University, Gorakhpur (U.P.)-273009,India) E-mail: profhsshuklagkp@rediffmail.com, [email protected]

-change of the Finsler metric given by Abstract: Abstract: We have considered the conformal β -change ¯ (x, y ) = eσ(x) f (L(x, y ), β (x, y)), L(x, y ) → L )), where σ (x) is a function of x, β (x, y ) = bi (x)y i is a 1-form on the underlying manifold M n ,and f (L(x, y ), β (x, y )) is a homogeneous function of degree one in L and β .We .We have studied quasi-C-reducibility, C-reducibility and semi-C-reducibility of the Finsler space with this metric. We have also calculated V-curvature tensor and T-tensor of the space with this changed metric in terms of v-curvature tensor and T-tensor respectively of the space with the original metric.

Key Words ords:

Conforma Conformall change, change, β -change, -change, Finsler Finsler space, space, quasi-C-r quasi-C-reduc educibili ibility ty,, C-

reducibility reducibility, semi-C-reducibility semi-C-reducibility,, V-curvature V-curvature tensor, T-tensor.

AMS(2010): AMS(2010): 53B40, 53C60. §1. Introduction

Let F n = (M n , L) be an n-dim n-dimens ension ional al Finsle Finslerr space space on the differe different ntial ialble ble manif manifold old M n equipped with the fundamental function L(x,y).B.N.Prasad and Bindu Kumari and C. Shibata [1,2] have studied the general case of β of β -change,that -change,that is, L ∗ (x, y ) = f ( f (L, β ),where ),where f is positively homogeneous function of degree one in L and β , and β and β given given by β by β (x, y ) = b i (x)y i is a one- form on M n . The The β -change -change of special Finsler spaces has been studied by H.S.Shukla, O.P.Pandey and Khageshwar Mandal [7]. The conformal theory of Finsler space was initiated by M.S. Knebelman [12] in 1929 and has been investigated in detail by many authors (Hashiguchi [8] ,Izumi[4,5] and Kitayama [9]). The conformal change is defined as L ∗ (x, y ) = e σ(x) L(x, y), where σ where σ((x) is a function of position ¯ (x, y ) = only and known as conformal factor. factor. In 2008, Abed [15,16] [15,16] introduced introduced the change L eσ(x) L(x, y ) + β ( β (x, y ), whic which h he call called ed a β -confo -conforma rmall chang change, e, and in 2009 and 201 2010,N 0,Nabi abill σ L.Youssef, L.Y oussef, S.H.Abed and S.G. Elgendi [13,14] introduced the transformation ¯ L(x, y ) = f ( f (e L, β ), ), which is β is β -change -change of conformally changed Finsler metric L. They have not only established the relationships between some important tensors of ( M n , L) and the corresponding tensors of (M n , ¯ L), but have have also studied several properties properties of this change. 1 Received

January 22, 2018, Accepted May 12, 2018.

14

H.S.Shukla H.S.Shukla and Neelam Mishra

We have changed the order of combination of the above two changes in our paper [6], where we have applied β applied β -change -change first and conformal change afterwards, i.e., ¯ (x, y ) = eσ(x) f ( L f (L(x, y ), β (x, y )), )),

(1.1)

where σ (x) is a function of x, β (x, y ) = bi (x)y i is a 1-form 1-form.. We have have called called this change change as conformal β conformal β -cha -change nge of Finsler Finsler metric. metric. In this paper we have have investig investigated ated the condition under which a conformal β -change -change of Finsler metric leads a Douglas space into a Douglas space.We have also found the necessary and sufficient conditions for this change to be a projective change. In the present paper,we investigate some properties of conformal β -change. -change. The Finsler Finsler ¯ given by (1.1) will be denoted by F ¯n .Throughout the paper space equipped with the metric L the quantiti quantities es correspondin correspondingg to F ¯n will be denoted by putting bar on the top of them.We ˙i respectively. shall denote the partial derivatives with respect to xi and yi by ∂ i and ∂ respectively. The n Fundamental quantities of F of F are given by 2

˙i ˙∂ j L = h ij + li lj , li = ∂ ˙ iL. gij = ∂ 2 Homogeneity of f f gives Lf 1 + βf 2 = f = f,,

(1.2)

where subscripts 1 and 2 denote the partial derivatives with respect to L and β β respectively. Differentiating above equations with respect to L and β respectively, β respectively, we get Lf 12 12 + βf 22 22 = 0 and Lf 11 11 + βf 21 21 = 0.

(1. (1.3)

Hence we have 2 2 f 11 = f 22 11 /β = ( f 12 12 )/Lβ = f 22 /L ,

−

(1.4)

which gives 2 f 11 11 = β ω, f 12 12 =

−Lβω,f 2222 = L = L2 ω,

(1.5)

where Weierstrass function ω is positively homogeneous of degree -3 in L and β . β . Therefore 3 ω = 0, Lω1 + βω 2 + 3ω

(1.6)

where ω1 and ω2 are positively homogeneous of degree -4 in L and β . Throughout Throughout the the paper we frequently frequently use the above equations equations without without quoting them. Also we have have assumed assumed that f is not linear function of L and β so that ω = 0. β so that ω



The concept of concurrent vector field has been given by Matsumoto and K. Eguchi [11] and S. Tachibana [17], which is defind as follows: The vector field b i is said to be a concurrent vector field if bi|j =

−gij

bi

|j = 0,0 ,

(1.7)

where small and long solidus denote the h- and v-covariant derivatives respectively. It has been

15

Some Properties of Conformal β Conformal β -Change

proved by Matsumoto that b i and its contravariant components b i are functions of coordinates i alone. Therefore from the second equation of (1.7),we have C ijk ijk b = 0. The aim of this this paper paper is to study study some some special special Finsler Finsler spaces spaces arising arising from from confor conformal mal β -change -change of Finsler Finsler metric,vi metric,viz., z., quasi-C-re quasi-C-reducib ducible, le, C-reducibl C-reduciblee and semi-C-red semi-C-reducib ucible le Finsler Finsler spaces. spaces. Further, urther, we shall obtain v-curvature v-curvature tensor tensor and T-tensor T-tensor of this space and connect them with v-curvature tensor and T-tensor respectively of the original space.

¯n Tensor and Angular Angular Metric Tensor Tensor of F §2. Metric Tensor Differentiating equation (1.1) with respect to y i we have ¯li = e = e σ (f 1 li + f 2 bi).

(2.1)

Differentiating (2.1) with respect to y j , we get ¯ ij = e h

where m where m i = b i

2σ





f f 1 hij + f L2 ωm imj , L

(2.2)

− Lβ Li .

¯n: From (2.1) and (2.2) we get the following relation between metric tensors of F n and F g¯ij = e where p where p = = f f 1 f 2

2σ



f f 1 gij L

−



pβ l i lj + (f ( f L2 ω + f 22 )bi bj + p + p((bi lj + bj li ) , L

(2.3)

− fβLω. fβLω.

¯ n , obtainable from ¯g ij g¯jk = δ i , The contravariant components ¯g ij of the metric tensor of F k are as follows: ij

g¯ = e

−2σ



L ij pL3 g + 3 f f 1 f f 1 t



f β L2

− ∆f 2



i j

ll

−

L4 ω i j bb f f 1t

−

pL2 i j (l b + lj bi) , 2 f f 1 t



(2.4)

where l where l i = g ij lj , b 2 = b i bi , b i = g ij bj , g ij is the reciprocal tensor of g of g ij of F F n , and t = f = f 1 + L3 ω∆, ∆ = b = b 2





2

− Lβ 2 .

f ˙i f = e ˙i f 1 = eσ βLωmi , (a) ∂ li + f 2 mi , (b) ∂ L ˙i f 2 = e ˙i p = (c)∂ = e σ L2 ωmi , (d) ∂ p = βqLmi , 3ω ˙i b2 = 2C ..i (e) ∂ ˙i ω = li + ω2 mi , (f ) f )∂ ..i , L 2β 2 β ˙i ∆ = 2C ..i (g ) ∂ mi , ..i L2 σ

− −

(2.5)

−

−

−

−

(2. (2.6)

16


− 3Lq li , (b) ∂ ˙i t = −2L3ωC ..i..i + [L [ L3 ∆ω2 − 3βLω] βLω ]mi, 3q ˙i q = (c) ∂ = − l i + (4f (4 f 2 ω2 + 3ω 3 ω2 L2 + f ω22 )mi . L ˙i q = (a) ∂ =

(2. (2.7)

¯n §3. Cartan’s C-Tensor and C-Vectors C-Vectors of F Cartan’s covariant C-tensor C ijk F n is defined by ijk of F 1 ˙ ˙ ˙ 2 ¯ijk C ∂ i ∂ j ∂ k L = ∂ ˙k gij ijk = 4 and Cartan’s C-vectors are defined as follows: jk i i jk C i = C ijk ijk g , C = C jk g .

(3.1)

We shall write C 2 = C i C i . Under Under the conform conformal al β -chang -chang (1.1) we get the following relation n n ¯ : between Cartan’s C-tensors of F of F and F p qL q L2 2σ f f 1 ¯ijk C C ijk (hij mk + hjk mi + hki mj ) + mi mj mk . ijk = e ijk + L 2L 2





(3.2)

We have (a) m i li = 0, 0, β 2 (b) m i b = b = ∆ = b imi , 2 L i (c) g ij m = h ij mi = m j . i

2

−

(3. (3.3)

From (2.1), (2.3), (2.4) and (3.2), we get h ¯ij C

p qL 3 (hij mh + hhj mi + hhi mj ) + mj mk mh 2f f 1 2f f 1 L pL∆ pL∆ 2 pL 2 pL + qL 4 ∆ h C.jk nh h n mj mk nh , jk ft 2f 2 f 1 t 2f 2 f 1 t

h = C ij +

−

−

−

(3. (3.4)

r r j where n where n h = f L2 ωb h + plh and h and h ji = g il hlj , C .ij .ij = C rij rij b , C ..i ..i = C rji rji b b and so on.

¯ n = (M n , ¯ n-dimensiona ionall Finsler Finsler spac spacee obtaine obtained d from from the Proposition 3.1 Let F L) be an n-dimens n n conformal β β -change of the Finsler space F F = (M ( M , L), then the normalized supporting element ¯li, angular metric tensor h ¯ ij , fundamental metric tensor g¯ij and (h)hv-torsion tensor C ¯ijk ijk of n ¯ are given by (2.1), (2.2), (2.3) and (3.2), respectively. F From (2.4),(3.1),(3.2) and (3.4) we get the following relations between the C-vectors of of ¯ n and their magnitudes F and F n

¯i = C C = C i

− L3ωC i..i.. + µmi,

(3.5)

17


where

p(n + 1) µ = 2f f 1

−

3 pL 3 pL3 ω ∆ qL q L3 ∆(1 L3 ω ∆) + ; 2f f 1 2f f 1

−

−2σ L i ¯i = e C C + M i , f f 1

where

µe−2σ L i M = m f f 1 i

−

L 4 ω i C f f 1 ..

and

−

2σ

− e

C i



3

L ωC i.. i.. + µ∆

−2σ ¯2 = e C C 2 + λ, p

(3.6)

L3 ω i L b + yi f f 1 ft



(3.7)

where λ =



e−2σ L f f 1

3



− L ω∆

µ2 ∆ +

2µe 2 µe−2σ L C . f f 1

− (1 + 2µ 2 µ∆) L3 ω + 1 − 3µ + e2σ L2 ωf f 1C . L3 ωC ... ... 3 r 2σ 2 r r +L3 ωC ..r e4σ Lωf 2 f 12 C i.. ..r i.. − µ∆ L ωb − e L ωf f 1 C .. − 2C











.

¯n §4. Special Cases of F

¯ n which is In this section, following Matsumoto [10], we shall investigate special cases of F conformally β conformally β -changed -changed Finsler space obtained from F n .

Definition 4.1 A Finsler space (M n , L) with dimension n if the Cartan tensor C ijk ijk satisfies

≥ 3 is said to be quasi-C-reducible

C ijk ijk = Q ij C k + Qjk C i + Qki C j ,

(4.1)

where Qij is a symmetric indicatory tensor.

The equation (3.2) can be put as



1 2σ f f 1 ¯ijk C C ijk ijk = e ijk + π(ijk) ijk ) L 6



 

3 p h ij + qL 2 mimj mk L

,

where π where π (ijk) and k.. ijk ) represents cyclic permutation and sum over the indices i, j and k Putting the value of m of m k from equation (3.5) in the above equation, we get



1 2σ f f 1 ¯ijk C C ijk π(ijk) ijk = e ijk + ijk ) L 6µ



3 p ¯k h ij + qL 2 mi mj )(C L

3

− C k + L ωC k.. k..



.

18


Rearranging this equation, we get ¯ijk C ijk





    −

f f 1 1 3 p ¯k = e2σ C ijk π(ijk) h ij + qL 2 mi mj C ijk + ijk) L 6µ L 1 3 p + π(ijk) h ij + qL 2 mi mj L3ωC k.. C k k.. ijk ) 6µ L





.

Further rearrangment of this equations gives ¯ijk ¯ ij ¯ C C k ) + U ijk ijk = π (ijk) ijk , ijk ) (H ij ¯ ij where H ij =

e2σ 6µ

(4.2)

{( 3L p hij + qL2mimj ), and

U ijk ijk = e

2σ



f f 1 1 C ijk π(ijk) ijk + ijk) L 6µ



3 p h ij + qL 2 mi mj L



3

L ωC k.. k..

− C k



(4.3)

¯ ij Since H ij is a symmetric and indicatory tensor,therefore from equation (4.2) we have the following theorem. ¯ n is quasi-C-reducible iff the tensor U ijk -changed Finsler space space F Theorem 4 Theorem 4..1 Conformally β -changed ijk of equation (4.3) vanishes identically. We obtain a generalized form of Matsumoto’s Matsumoto’s result [10] as a corollary of the above theorem. ¯n -changed Finsler space F Corollary 4.1 If F n is Reimannian space, then the conformally β -changed is always a quasi-C-reducible Finsler space. Definition 4 Definition 4..2 A Finsler space (M ( M n , L) of dimension n tensor C ijk ijk is written in the form C ijk ijk =

≥ 3 is called C-reducible if the Cartan

1 (hij C k + hki C j + hjk C i ). n+1

(4.4)

1 Define the tensor Gijk = C ijk ijk (n+1) (hij C k + h ki C j + h jk C i ). It is clear that Gijk is symmetric and indicatory. indicatory. Moreover, Moreover, G ijk vanishes iff F n is C-reducible.

−

¯ ijk associated with the space -change(1.1), the tensor G Proposition 4 Proposition 4..1 Under the conformal β -change(1.1), ¯ n has the form F ¯ ijk = e 2σ f f 1 Gijk + V ijk G (4.5) ijk L where V ijk ijk

=

1 2σ π(ijk) 1)(α1 hij + α2 mi mj )mk + e2σ ωL 2 mi mj C k ijk ) (e (n + 1)(α (n + 1)

{

+e2σ L2 ω(f f 1 hij + L3 ωmi mj )C k.. k.. ,

}

α1 =

e2σ p 2L

2σ

µf f 1e − Lµff , (n + 1)

α2 =

(4. (4.6) e2σ qL 2 6

2σ

2

− µe(n +ωL1) .

19


From (4.5) we have the following theorem. ¯ n is C-reducible iff F n is C-reducible -changed Finsler space F Theorem 4.2 Conformally β -changed and the tensor V ijk ijk given by (4.6) vanishes identically. Definition 4.3 A Finsler space (M n , L) of dimension n Cartan tensor C ijk ijk is expressible in the form:

≥ 3 is called semi-C-reducible if the

r s (hij C k + hki C j + hjk C i ) + 2 C i C j C k , n+1 C

C ijk ijk =

(4.7)

where r and s are scalar functions such that r + r + s = 1. Using equations (2.2), (3.5) and (3.7) in equation (3.2), we have ¯ijk C ijk = e

2σ





f f 1 p ∆L ∆ L(f 1 q 3 pω) pω ) ¯ ¯ ¯ ¯ ij ¯ C ijk (h C k + ¯hki ¯ C j + ¯hjk ¯ C i ) + C i C j C k . ijk + ¯ L 2µf f 1 2f f 1 µtC 2

−

If we put p(n + 1) ′ ∆L(f 1 q 3 pω) pω ) ,s = , 2µff µf f 1 2f f 1µt

−

r′ = we find that r that r ′ + s′ = 1 and

r′ ¯ ¯ s′ ¯ ¯ ¯ 2σ f f 1 ¯ijk ¯ ¯ ¯ ¯ C = e C + ( h C + h C + h C ) + ijk ijk ijk ij k ki j jk i ¯ 2 C i C j C k . L n+1 C





(4.8)

¯ n is semi-C- reducible iff C ijk From equation (4.8) we infer that F i.e.. iff F n is a ijk = 0, i.e Reimannian space. Thus we have the following theorem. ¯ n is semi-C-reducible iff F n is a Rie-changed Finsler space F Theorem 4.3 Conformally β -changed mannian space. ¯n §5. v-Curvature Tensor of F The v The v-curvature -curvature tensor [10] of Finsler space with fundamental function L is given by r S hijk hijk = C ijr ijr C hk

r − C ikr ikr C hj

¯ n will be given by Therefore the v the v -curvature tensor of conformally β -changed β -changed Finsler space F r ¯hijk ¯ijr ¯ S C hk hijk = C ijr

¯ r − C ¯ikr ikr C hj .

From equations (3.2)and(3.4), we have ¯ijr ¯ r C ijr C hk



f f 1 p r C ijr (C ijk ijr C hk + ijk mh + C ijh ijh mk + C ihk ihk mj L 2L pf 1 f f 1 L2 ω +C hjk (C .ij C .ij hjk mi ) + .ij hhk + C hk hk hij ) .ij C .hk .hk 2Lt t 2σ

= e

−

(5.1)

20


p 2 ∆ L 2 (qf 1 2 pω) pω ) hhk hij + (C .ij .ij mk mh + C .hk .hk mi mj ) 4f Lt 2t p( p( p + L3 q ∆) ∆) p2 + (hij mh mk + hhk mi mj ) + (hij mh mk 4Lf t 4Lf f 1 +hhk mi mj + hhj mi mk + hhi mj mk + hjk mi mh + hik mh mj ) L2 (2 pqt + (qf (qf 1 2 pω)(2 pω )(2 p + L3 q ∆)) ∆)) + mi mj mh mk . 4f f 1 t

−

+



−

(5. (5.2)

We get the following relation between v-curvature tensors of ( M n , L) and (M (M n , ¯ L):



2σ f f 1 ¯hijk S S hijk hijk = e hijk + dhj dik L

− dhkdij + E hk hk E ij ij − E hj hj E ik ik



,

(5.3)

where

− Qhij + Rmimj ,

dij = P C .ij .ij

E ij S hij + T mimj , ij = Sh

P = L

s t

1/2



, Q =

(5.4)

√ −

(5.5)

√ −

pg L (2ωp (2ωp f 1 q ) p L (qf 1 ωp) ωp) , R = , S = = , T = . 2 2 2L f ω 2f 1 f ω 2L st 2 st

√

√

¯ n is given by (5.3). between v-curvature tensors of F n and F Proposition 5 Proposition 5..1 The relation between When b When b i in β in β is is a concurrent vector field,then C .ij 0 . Therefore the value of v-curvature .ij = 0. n ¯ tensor of F as given by (5.3) is reduced to the extent that d ij = Rm i mj Qhij .

−

§6. The T-Tensor T hijk hijk

The T-tensor of F F n is defined in [3] by T hijk hijk = LC hij hij

|k +C + C hij hij lk + C hik hik lj + C hjk hjk li + C ijk ijk lh ,

(6.1)

where C hij hij

r r r |k = ∂ ˙k C hij hij − C rij rij C hk − C hrj hrj C ik − C hir hir C jk .

(6.2)

¯ n , which is given by In this section we compute the T-tensor of F ¯hijk ¯ ¯hij¯ + C ¯hij ¯ ¯hik l¯j + C ¯hjk ¯ ¯ijk ¯lh , T hijk = LC hij hij lk + C hik hjk li + C ijk k

|

(6.3)

where ¯ ˙ ¯hij ¯hij C hij k = ∂ k C hij

|

r ¯ r ¯hrj ¯ ¯ r − C ¯rij − C ¯hir C ik rij C hk − C hrj hir C jk .

(6.4)

The derivatives of m of m i and h and h ij with respect to y k are given by ˙k mi = ∂

1 − β hik − (li mk ), 2 L L

˙k hij = 2C ∂ 2 C ijk ijk

− L1 (lihjk + lj hki)

(6.5)

21


From equations (3.2)and (6.5), we have ∂ ˙k ¯ C hij hij



f f 1 p ∂ k C hij (C ijk hij + ijk mh + C ijh ijh mk + C ihk ihk mj + C hjk hjk mi ) L L pβ p (hij hhk + hhj hik + hih hjk ) + (hjk lh mi + hhk lj mi 3 2L 2L2 +hhk li mj + hik lh mj + hjk li mh + hjk lh mi + hij lhmk + hhj li mk β q +hik lj mk + hij lk mh + hjh lk mi + hhi lk mj ) (hij mh mk 2 +hhk mi mj + hhj mi mk + hhi mj mk + hjk mimh + hik mh mj ) qL (li mj mh mk + lj mi mh mk + lhmi mj mk + hk mi mj mh ) 2 L2 + (4f (4f 2 ω2 + 3L 3 L2 ω2 + f ω22 )mh mi mj mk . 2

= e

2σ

−

−

−



(6. (6.6)

Using equations (6.5) and (5.2) in equation(6.4), we get ¯ ¯hij C hij k

|

e2σ p = e C hij (C ijk + C hjk hij k ijk mh + C ijh ijh mk + C ihk ihk mj + C hjk mi ) L 2L 2f βt L2 p∆ p∆ βq 2σ 2σ pe + ( h h + h h + h h ) e ij hk hj ik ih jk 4f L 3 t 4f L 3 t 2 2 3 2 p f 1 + pqf + pqf 1 L ∆ + 3 p 3 p + (hij mh mk + hhk mi mj + hhj mimk + hhi mj mk 4Lf f 1t e 2σ p +hjk mi mh + hik mh mj ) [l [ lh (hjk mi + hij mk 2L2 +hik mj ) + lj (hhk mi + hik mkh + hih mk ) + li (hhk mj + hjk mh e 2σ qL +hhj mk ) + lk (hij mh + hjh mi + hhi mj )] (li mj mh mk 2 pf 1 e2σ +lj mi mh mk + lh mi mj mk + hk mi mj mh ) (C .ij .ij hhk 2Lt e 2σ f f 1 L2 ω +C .hj (C .ij .hj hik + C .hk .hk hij + C .ik .ik hh + C .hi .hi hjk + C .jk .jk hhi ) + .ij C .hk .hk t e 2σ L2 (qf 1 2 pω) pω ) +C .hj C + C C ) (C .ij .hj .ik .ik .hi .hi .jk .jk .ij mk mh 2t +C .hk .hk mi mj + C .hj .hj mi mk + C .ik .ik mj mh 2 (4f 2 ω2 + 3L 3 L2 ω2 + f ω22 ) 2σ L (4f +C .hi m m + C m m ) + e .hi j k .jk .jk h i 2 3L2 (2 pqt + (qf (qf 1 2 pω)(2 pω )(2 p + L3 q ∆) ∆) mi mj mh mk . (6. (6.7) 4f f 1 t 2σ f f 1

−

| −



 

−



−

− −

−

−

−

−





Using Using equati equations ons (2.1), (2.1), (3.2) (3.2) and (6.6) (6.6) in equati equation on (6.3), (6.3), we get the follo followin wingg relati relation on

22


¯n : between T-tensors of Finsler spaces F n and F ¯hijk T hijk

f 2 f 1 f ( f (f 1 f 2 + fβLω) fβLω ) T hijk (C ijk hijk + ijk mh + C ijh ijh mk + C ihk ihk mj 2 L 2L f 2 f 1 L2 ω +C hjk (C .ij hjk mi ) + .ij C .hk .hk + C .hj .hj C .ik .ik + C .hi .hi C .jk .jk ) t pf 1 (C .ij .ij hhk + C .hj .hj hik + C .hk .hk hij + C .ik .ik hh + C .hi .hi hjk + C .jk .jk hhi ) 2Lt f L2 (qf 1 2 pω) pω ) (C .ij .ij mk mh + C .hk .hk mi mj + C .hj .hj mi mk 2t p(2 p (2ff βt + L2 p∆) p∆) +C .ik (hij hhk .ik mj mh + C .hi .hi mj mk + C .jk .jk mh mi ) 3 4L t p2 f 1 + pqf + pqf 1 L3 ∆ + 3 p 3 p2 β qf pf 2 +hhj hik + hih hjk ) + 4Lf 1t 2 L (hij mh mk + hhk mi mj + hhj mi mk + hhi mj mk + hjk mimh L2 (4f (4f 2 ω2 + 3L 3 L2 ω2 + f ω22 ) +hik mh mj ) + + 2L 2L2 f 2 q 2 2 3L (2 pqt + (qf (qf 1 2 pω)(2 pω)(2 p + L3 q ∆) ∆) mi mj mh mk . 4f 1 t 3σ

= e



− −

−

−

 −



−

−



−





(6. (6.8)

¯ n is given by (6.7). Proposition 6 Proposition 6..1 The relation between T-tensors of F n and F If bi bi is a concurrent vector field in F n , then C then C .ij .ij = 0. Therefore from(6.8), we have ¯hijk T hijk

f 2 f 1 p(2 p (2ff βt + L2 p∆) p∆) T (hij hhk + hhj hik + hih hjk ) hijk h ijk 2 3 L 4L t p2 f 1 + pqf + pqf 1 L3 ∆ + 3 p 3 p2 t β qf pf 2 + (hij mh mk + hhk mi mj 4Lf 1 t 2 L +hhj mi mk + hhi mj mk + hjk mi mh + hik mh mj ) L 2 (4f (4f 2 ω2 + 3L 3 L2 ω2 + f ω22 ) 3L 3 L2 (qf 1 2 pω)(2 pω )(2 p + L3 q ∆) ∆) + 2L2 f 2 q + + + 2 4Lf f 1t 2 3L 2 pqt mi mj mh mk . 4Lf f 1t

  −  3σ

= e

−

−

−



−





(6. (6.9)

If bi is a concurrent vector field in F n , with vanishing T-tensor then T-tensor of F n is given by ¯hijk T hijk

p(2 p(2ff βt + L2 p∆) p∆) = e (hij hhk + hhj hik + hih hjk ) 3 4L t p2 f 1 + pqf + pqf 1 L3 ∆ + 3 p 3 p2 t β qf pf 2 + (hij mh mk 4Lf 1 t 2 L +hhk mi mj + hhj mi mk + hhi mj mk + hjk mi mh + hik mh mj )

−  −  3σ

−



L2 (4f (4f 2 ω2 + 3L 3 L2 ω2 + f ω22 ) 3L 3 L2 2 pqt 2 4Lf f 1t 3L2(qf 1 2 pω)(2 pω )(2 p + L3 q ∆) ∆) + + 2L 2L2 f 2 q mi mj mh mk . 4Lf f 1t

+

−

−





(6. (6.10)


23

Acknowledgement The work contained in this research paper is part of Major Research Project “Certain Investigations in Finsler Geometry” financed by the U.G.C., New Delhi.

References [1] B.N.Prasad B.N.Prasad and Bindu Kumari, Kumari, The β -change β -change of Finsler metric and imbedding classes of their tangent spaces, Tensor N. S., 74, (2013),48-59. Math.Kyoto to Univ. Univ. , [2] C. Shibat Shibata, a, On inv invarian ariantt tensors tensors of β -change -change of Finsler Finsler metric, metric, J. Math.Kyo 24(1984),163-188. [3] F.Ikeda, F.Ikeda, On the tensor T tensor T ijkl ijkl of Finsler spaces, Tensor N. S. , 33(1979),203-209. [4] H.Izumi, Conformal transformations transformations of Finsler spaces I, Tensor N.S., 31(1977),33-41. [5] H.Izumi, H.Izumi, Conformal Conformal transformati transformations ons of Finsler Finsler spaces II. An h-conforma h-conformally lly flat Finsler Finsler space, Tensor N.S., 33(1980),337-369. [6] H.S.Shukla and Neelam Mishra, Mishra, Conformal β -changes -changes of Finsler Finsler metric, metric, J.Int.Acad.Phys.Sci. , 21(1)(2017), 19-30. [7] H.S.Shukla, O.P.pandey O.P.pandey and Khageshwar Mandal, The β The β -change -change of Special Finsler Spaces, International International J.Math.Combin. J.Math.Combin., 1(2017), 78-87. [8] M.Hashiguchi, On conformal transformation of Finsler metric, J.Math.Kyoto Univ. ,16(1976),2550. Geometry of Transfo ransformat rmations ions of Finsler Finsler metrics metrics , Hokkaido [9] M.Kitayama, M.Kitayama, Geometry Hokkaido University University of Education,Kushiro Campus, Japan, 2000. Foundationss of Finsler Geometry and Special Special Finsler Spaces Spaces , Kaiseisha [10] M.Mastsumoto, Foundation Press, Saikawa,Otsu, 520 Japan, 1986. [11] M. Matsumoto Matsumoto and K. Eguchi, Eguchi, Finsler space admittin admittingg a concurren concurrentt vector field, Tensor N.S., 28(1974), 239-249. Proc.Nat.Acad cad.. Sci., [12] M.S.Knebel M.S.Knebelman, man, Conformal Conformal geometry of generalize generalized d metric metric spaces, spaces, Proc.Nat.A USA.,15(1929),376-379. [13] N.L.Youssef, N.L.Youssef, S.H.Abed and S.G.Elgendi, Generalized β -conformal β -conformal change of Finsler metrics, Int.J.Geom.Meth.Mod.Phys. ,7,4(2010).ArXiv No.:math.DG/0906.5369. [14] N.L.Youssef,S.H.Abed N.L.Youssef,S.H.Abed and S.G.Elgendi, S.G.Elgendi, Generalized β -conformal -conformal change change and special Finsler Finsler spaces, Int.J.Geom.Meth.Mod.Phys. , 8,6(2010). ArXiv No.:math.DG/1004.5478v3. [15] S.H.Abed, Conformal β Conformal β -changes -changes in Finsler spaces, Proc.Math.Phys.Soc.Egypt , 86(2008),7989.ArXiv No.:math.DG/0602404. [16] S.H.Abed, S.H.Abed, Cartan connections connections associated with a β -conformal -conformal change in Finsler geometry, Tensor N.S., 70(2008),146-158. ArXiv No.:math.DG/0701491. [17] S.Tachi S.Tachibana, bana, On Finsler Finsler spaces spaces which which admit admit a concurren concurrentt vector vector field, Tensor N. S. , 1 (1950),1-5.


Equitable Coloring on Triple Star Graph Families K.Praveena (Department of Computer Science, Dr.G.R. Damodaran College of Science, Coimbatore-641014, Tamilnadu, India)

M.Venkatachalam (Department of Mathematics, Kongunadu Arts and Science College, Coimbatore-641029, Tamilnadu, India) E-mail: [email protected], [email protected] [email protected]

Abstract: Abstract: An equitable k-coloring of a graph G is a proper k-coloring of G such that the sizes sizes of any any two two color color class class differ differ by at most one. In this paper we investi investigat gatee the equitable equitable chromatic chromatic number for the Central Central graph, graph, Middle Middle graph, graph, Total graph and Line graph of Triple star graph K 1,n,n,n denoted by C (K 1,n,n,n ), M (K 1,n,n,n ), T ( T (K 1,n,n,n ) and L(K 1,n,n,n ) respectively.

Key Words: Words: Equitable coloring, Smarandachely equitable k-coloring, triple star graph, central graph, middle graph, total graph and line graph.

AMS(2010): AMS(2010): 05C15, 05C78. §1. Introduction

A graph consist of a vertex set V ( V (G) and an edge set E set E (G). All Graphs in this paper are finite, finite, loopless loopless and without without multiple multiple edges. We refer refer the reader [8] for terminolog terminology y in graph theory. theory. Graph coloring coloring is an important important research research problem problem [7, 10]. A proper k -coloring of a graph is a labelling f : V ( V (G) 1, 2, , k such that the adjacent adjacent vertices vertices have different different labels. labels. The labels are colors and the vertices vertices with same color form a color class. The chromatic chromatic number number of a graph G graph G,, written as χ( χ (G) is the least k such that G that G has a proper k-coloring. k -coloring. Equitable Equitable colorings naturally naturally arise in some schedulin scheduling, g, partitioni partitioning ng and load balancing problems problems [11,12]. [11,12]. In 1973, Meyer [4] introduced introduced first the notion notion of equitable equitable colorabilit colorability y. In 1998, Lih [5] surveyed the progress on the equitable coloring of graphs. We say that a graph G = (V, E ) is equitably k-colorable if and only if its vertex set can be partitioned into independent sets V 1 , V 2 , , V k V V such that V i V j 1 holds for every pair (i, (i, j ). The smallest smallest integer integer k for which G is equitable k -colorable is known as the equitable chromatic number [1,3] of G and denoted by χ= (G). On the other other hand, hand, if V can be partitioned into independent sets V 1 , V 2 , , V k V V with V i V j 1 holds for every pair (i, (i, j ), such a k-coloring k -coloring is called a Smarandachely equitable k -coloring . In this paper, we find the equitable chromatic number χ = (G) for central, line, middle and total graphs of triple star graph.

→ {

1 Received

··· }

{

···

}⊂

{

···

}⊂

September 14, 2017, Accepted May 15, 2018.

|| | − | || ≤

|| | − | || ≥

25

Equitable Coloring on Triple Star Graph Families

§2. Preliminaries

For a given graph G graph G = (V, E ) we do a operation on G, G , by subdividing each edge exactly once and joining all the non adjacent vertices of G. The graph graph obtain obtained ed by this this process process is called called central graph of G of G [1] and is denoted by C ( C (G). The line graph [6] [6] of a graph G, G , denoted by L(G) is a graph whose vertices are the edges of G G and if u, u, v E (G) then uv then uv E (L(G)) if u u and v share a vertex in G. G .

∈

∈

Let G be a graph with vertex set V ( V (G) and edge set E (G). The The middle graph [2] of G denoted by M by M ((G) is defined as follows. The vertex set of M of M ((G) is V is V ((G) E (G) in which two vertices x, y are adjacent in M ( M (G) if the following condition hold:

∪

(1) x, (1) x, y

∈ E (G) and x, and x, y are adjacent in G; G ; (2) x (2) x ∈ V ( V (G), y ∈ E (G) and they are incident in G. G . Let G be a graph with vertex set V ( V (G) and edge set E (G). The The total graph [1,2] of G is denoted by T ( T (G) and is defined defined as follow follows. s. The vertex vertex set of T ( T (G) is V ( V (G) E (G). Two Two vertices x, y in the vertex set of T of T ((G) is adjacent in T ( T (G), if one of the following holds:

∪

(1) x, (1) x, y are in V in V ((G) and x and x is adjacent to y in G in G;; (2) x, (2) x, y are in E in E ((G) and x, and x, y are adjacent in G; G ; (3) x (3) x is in V ( V (G), y ), y is in E in E ((G) and x, and x, y are adjacent in G. G . Triple star K star K 1,n,n,n [9] is a tree obtained from the double star [2] K 1,n,n by adding a new pendant edge of the existing n pendant vertices. It has 3n 3 n + 1 vertices and 3n 3 n edges.

§3. Equitable Equitable Coloring on Central Central Graph of Triple Star Graph

Algorithm 1. Input: The number number ‘n’ ‘n’ of k k 1,n,n,n ; Output: Assigning equitable colouring colouring for the vetices vetices in C ( C (K 1,n,n,n ).

begin for i for i = 1 to n

{

{ }

V 1 = ei

C (ei ) = i; i ; V 2 = ai ;

{ }

C (ai ) = i; i;

} V 3 = v ;

{ }

C (v) = n + 1;

26

K.Praveena and M.Venkatachalam

for i for i = = 2 to n

{ V 4 = vi ;

{ }

− 1; V 5 = {wi }; C (wi ) = i − 1; V 6 = {ui}; C (ui ) = i = i − 1; } C (vi ) = i

C (v1 ) = n; n ; C (w1 ) = n; n ; C (u1 ) = n; n; for i for i = = 1 to 5

{

{ }

V 7 = si ; C (si ) = n + 1;

} for i for i = = 6 to n

{ V 8 = si ;

{ }

C (si ) = i; i ;

} V = V 1

∪ V 2 ∪ V 3 ∪ V 4 ∪ V 5 ∪ V 6 ∪ V 7 ∪ V 8;

end Theorem 3.1 For any triple star graph K 1,n,n,n the equitable chromatic number χ= [C (K 1,n,n,n )] = n + 1. 1.

Proof Let vi : 1 i n , wi : 1 i n and ui : 1 i n be the vertices in K 1,n,n,n . The vertex vertex v is adjacent to the vertices v i (1 i n). n ). The vertice verticess v i (1 i n) n ) is adjacent to the vertices w i (1 i n) n) and the vertices w i (1 i n) n) is adjacent to the vertices ui(1 i n). n ).

{

≤ ≤

≤ ≤ } { ≤ ≤

≤ ≤ }

{ ≤ ≤ } ≤ ≤ ≤ ≤

≤ ≤

≤ i ≤ n) n )

By the definition definition of central graph on K 1,n,n,n, let the edges vv v vi , vi wi and w and w i ui (1 of K K 1,n,n,n be subdivided by the vertices e i , ai , si (1 i n) n ) respectively. respectively.

≤ ≤

27


Clearly, V [ V [C (K 1,n,n,n )] =

The vertices v vertices v and u and u i (1 Therefore





{v} {vi : 1 ≤ i ≤ n} {wi : 1 ≤ i ≤ n} {ui : 1 ≤ i ≤ n} {ei : 1 ≤ i ≤ n} {ai : 1 ≤ i ≤ n} ∪ {si : 1 ≤ i ≤ n}

 



≤ i ≤ n) n) induces a clique of order n +1 (say k (say k n+1 ) in [C [ C [K 1,n,n,n ]]. ≥ n + 1

χ= [C (K 1,n,n,n )]

Now consider the vertex set V set V [[C (K 1,n,n,n )] and the color class C class C = = c1 , c2 , c3 , an equitable coloring to C ( C (K 1,n,n,n ) by Algorithm 1. Therefore

{

≤ n + 1.1.

χ= [C (K 1,n,n,n )]

|| | − |vj || ≤ 1. Hence

An easy check shows that vi

χ= [C (K 1,n,n,n )] = n + 1. 1.

Equitable Coloring on Line graph of Triple Star Graph §4. Equitable Algorithm 2. Input: Output:

The number number ‘n’ ‘n’ of of K K 1,n,n,n ; Assigning equitable equitable coloring for the vertices vertices in L in L((K 1,n,n,n ). begin for i for i = = 1 to n

{ V 1 = ei ;

{ }

C (ei ) = i; i ;

{ }

V 2 = si ; C (si ) = i; i ;

} for i for i = = 2 to n

{ V 3 = ai ;

{ }

C (ai ) = i

}

− 1;

· · · cn+1}. Assign

28


C (a1 ) = n; n ; V = V 1

∪ V 2 ∪ V 3 ;

end Theorem 4.1 For any triple star graph K 1,n,n,n the equitable chromatic number, χ= [L(K 1,n,n,n )] = n.

Proof Let vi : 1 i n , wi : 1 i n and ui : 1 i n be the vertices in K 1,n,n,n. The vertex v is adjacent to the vertices v i (1 i n) n ) with edges e i (1 i n). n ). The vertices v i (1 i n) n ) is adjacent to the vertices w i (1 i n) n ) with edges a edges a i (1 i n). n ). The vertices w i (1 i n) n) is adjacent to the vertices u i (1 i n) n ) with edges s i (1 i n). n ).

{

≤ ≤ } {

≤ ≤ }

≤ ≤ ≤ ≤

{ ≤ ≤ ≤ ≤ ≤ ≤

≤ ≤ }

By the definition of line graph on K 1,n,n,n the edges e edges e i , ai , si (1 vertices of L of L((K 1,n,n,n). Clearly

{



≤ ≤ ≤ ≤ ≤ ≤

≤ i ≤ n) n) of K K 1,n,n,n are the



≤ i ≤ n} {ai : 1 ≤ i ≤ n} {si : 1 ≤ i ≤ n}

V [ V [L(K 1,n,n,n )] = ei : 1

≤ i ≤ n) n) induces a clique of order n (say K (say K n ) in L in L((K 1,n,n,n ). Therefore

The vertices e vertices e i (1

χ= [L(K 1,n,n,n )]

≥ n. {

Now consider the vertex set V set V [[L(K 1,n,n,n)] and the color class C = C = c1 , c2 ,

· · · cn } .

Assign an equitable coloring to L( L (K 1,n,n,n ) by Algorithm 2. Therefore χ= [L(K 1,n,n,n )]

≤ n.

|| | − |vj || ≤ 1. Hence

An easy check shows that vi

χ= [L(K 1,n,n,n )] = n.

Equitable Coloring on Middle and Total Total Graphs of Triple Star Graph §5. Equitable

Algorithm 3. Input: Output:

The number number ‘n′ of K K 1,n,n,n ; Assigning equitable equitable coloring for the vertices vertices in M in M ((K 1,n,n,n ) and T and T ((K 1,n,n,n ).


begin for i for i = = 1 to n

{ V 1 = ei ;

{ }

C (ei ) = i = i;;

{ }

V 2 = si ; C (si ) = i; i ;

}

{}

V 3 = v ; C (v) = n + 1; for i for i = = 2 to n

{

{ }

V 4 = vi ; C (vi ) = i

− 1;

} C (v1 ) = n; n ; for i for i = = 3 to n

{ { }

V 5 = ai ; C (ai ) = i

− 2;

}

C (a1 ) = n + 1; C (a2 ) = n + 1; for i for i = = 4 to n

{ { }

V 6 = wi

C (wi ) = i

− 3;

} C (w1 ) = n

− 1;

C (w2 ) = n; n ;

29

30


C (w3 ) = n + n + 1; for i for i = = 1 to n

{ V 7 = ui ;

{ }

C (ui ) = i = i + 1;

} V = V 1

∪ V 2 ∪ V 3 ∪ V 4 ∪ V 5 ∪ V 6 ∪ V 7

end Theorem 5.1 For any triple star graph K 1,n,n,n the equitable chromatic number, χ= [M ( M (K 1,n,n,n )] = n + n + 1, 1, n

Proof Let V Let V ((K 1,n,n,n ) = v

≥ 4. 4 .





{ } {vi : 1 ≤ i ≤ n}∪ {wi : 1 ≤ i ≤ n} {ui : 1 ≤ i ≤ n}. By the definition of middle graph on K 1,n,n,n each edge v edge vvvi , v i wi and w and w i ui (1 ≤ i ≤ n) n ) in K 1,n,n,n are subdivided by the vertices e i , w i , s i (1 ≤ i ≤ n) n) respectively. Clearly V [ V [M (K 1,n,n,n )] =





{v} {vi : 1 ≤ i ≤ n} {wi : 1 ≤ i ≤ n} {ui : 1 ≤ i ≤ n} {ei : 1 ≤ i ≤ n} {ai : 1 ≤ i ≤ n} {si : 1 ≤ i ≤ n}

 

 

≤ i ≤ n) n ) induces a clique of order n +1 (say k (say kn+1 ) in [M [ M (K 1,n,n,n )].

The vertices v vertices v and and e ei (1 Therefore

≥ n + 1.1.

χ= [M ( M (K 1,n,n,n )]

Now consider the vertex set V [ V [M (K 1,n,n,n )] and the color class C = Assign an equitable equitable coloring to M ( M (K 1,n,n,n ) by Algorithm 3. Therefore χ= M [( M [(K K 1,n,n,n )]

{c1, c2, · · · cn+1}.

≤ n + 1,1, ||vi | − |vj || ≤ 1. 1 .

Hence

∀ ≥ 4. 4 .

χ= [M ( M (K 1,n,n,n )] = n + 1 n

Theorem 5.2 For any triple star graph K 1,n,n,n the equitable chromatic number, χ= [T ( T (K 1,n,n,n )] = n + 1, 1, n

 

Proof Let V Let V ((K 1,n,n,n ) = v vi : 1 i E (K 1,n,n,n ) = ei : 1 i n ai : 1 i n

{



≥ 4. 4 .



{ } { ≤ ≤ n } {wi : 1 ≤ i ≤ n } {ui : 1 ≤ i ≤ n } and a nd ≤ ≤ } { ≤ ≤ } ∪ {si : 1 ≤ i ≤ n}.

31


≤ ≤ n) of K 1,n,n,n be

By the definition of Total graph, the edge vvi , vi wi and wi ui (1 i subdivided by the vertices e i , a i and s and s i(1 i n) n) respectively. Clearly

≤ ≤

V [ V [T ( T (K 1,n,n,n )] =





{v} {vi : 1 ≤ i ≤ n} {wi : 1 ≤ i ≤ n} {ui : 1 ≤ i ≤ n} {ei : 1 ≤ i ≤ n} {ai : 1 ≤ i ≤ n} {si : 1 ≤ i ≤ n}.

 

 

≤ i ≤ n) n ) induces a clique of order n + 1 (say (say k n+1 ) in T in T ((K 1,n,n,n ).

The vertices v vertices v and e i(1 Therefore

χ= [T ( T (K 1,n,n,n )]

≥ n + 1,1, n ≥ 4. 4 .

Now consider the vertex set V ( V (T ( T (K 1,n,n,n )) and the color class C = c1 , c2 , Assign an equitable equitable coloring to T ( T (K 1,n,n,n ) by Algorithm 3. Therefore

{

· · · , cn+1}.

χ= [T ( T (K 1,n,n,n)]

≤ n + 1,1, n ≥ 4, 4 , ||vi | − |vj || ≤ 1. 1 .

Hence

∀ ≥ 4. 4 .

χ= [T ( T (K 1,n,n,n)] = n + n + 1, 1, n

References [1] Akbar Ali. M.M, Kaliraj.K Kaliraj.K and Vernold ernold Vivin.J, Vivin.J, On equitable equitable coloring coloring of central central graphs and total graphs, Electronic Notes in Discrete Mathematics , 33,(2009),1-6. [2] Venkatachalam enkatachalam .M, Vernold Vernold Vivin.J, Akbar Ali. M.M, Star Coloring and Equitable coloring on Star and double star graph families, Mathematical and Computational Models, Recent trends , ICMCM 2009, P.S.G College of Technology, Coimbatore, Narosa Publishing House, New Delhi, India, December (2009), 286-291. [3] B.L.Chen and K.W.Lih , Equitable coloring of trees, J.Combin. Theory Ser.B, Ser.B, 61 (1994),8387. [4] W. Meyer, Equitable Equitable coloring, coloring, Amer. Math. Monthly , 80 (1973) 920-922. [5] K.-W. K.-W. Lih, Lih, The equitabl equitablee colori coloring ng of graphs, graphs, in: D.-Z. D.-Z. Du, P.M. Pardalo Pardaloss (Eds.) (Eds.),, in: Handbook of Combinatorial Optimization , Vol.3, Kluwer Academic Publishers, 1998, pp. 543-566. [6] Frank Harary, Graph theory, Narosa Publishing Home, 1969. [7] Li J.W., Zhang Z.F., Chen X.E., Sun Y.R., A Note on adjacent strong edge coloring of K(n,m), Acte Mathematicae Application Sinica , 22(2): 273-276 (2006). [8] D.B. West, West, Introduction to Graph Theory (Second ed.), Prentice-Hall, Upper Saddle River, NJ, 2001. [9] Akhlak Mansuri, On harmonious chromatic chromatic number number of triple star graph, Journal of Hyperstructures , 5(1) (2016),26-32. [10] Chen X.E., Zhang Z.F., AVDTC AVDTC number of generalized Halin graphs with maximum maximum degree at least 6, Acte Mathematicae Application Sinica , 24(1): 55-58 (2008)

32


Domain Dec Decomp omposition; osition; Parallel Mul[11]] B. F. Smith, [11 Smith, P. E. Bjorstad Bjorstad and W. D. Gropp, Gropp, Domain tilevel tilevel Method Methodss for Elliptic Elliptic Partial Partial Differe Differentia ntiall Equation Equations, s, Cambridge University University Press, Cambridg Cambridge, e, 224 p. (1996). (1996). [12] A. Tucker, Tucker, Perfect graphs and an application to optimizing municipal services, SIAM Review , 15 (1973), 585-590.


On the Tangent Vector Fields of Striction Curves Along the Involute and Bertrandian Frenet Ruled Surfaces S ¸ eyda ey da Kılı¸ Kıl ı¸co˘ co˘glu glu (Faculty (Faculty of Education, De partment of Mathematics, Ba¸skent skent University, University, Ankara, Turkey)

S¨ uleyman uley man S ¸enyur ¸e nyurtt and Abduss Ab dussamet amet C ¸ alı¸skan skan (Faculty of Arts and Sciences, Department of Mathematics, Ordu University, Ordu, Turkey) E-mail: [email protected],sen [email protected],[email protected], [email protected], [email protected]

Abstract: Abstract: In this paper we consider nine special ruled surfaces associated to an involute of a curve α and its Bertrand mate α∗∗ with k1 =  0. They They are called as involute involute Frene Frenett ruled and Bertrandian Frenet ruled surfaces, because of their generators which are the Frenet vector fields of curve α curve α.. First we give the striction striction curves of all Frenet ruled surfaces. surfaces. Then the striction curves of involute and Bertrandian Frenet ruled surfaces are given in terms of the Frenet apparatus of the curve α. curve α. Some Some results are given on the striction curves of involute and Bertrand Frenet ruled surfaces based on the tangent vector fields in E 3 .

Key Words: Words: Frenet ruled surface, involute Frenet ruled surface, Bertrandian Frenet ruled surface, evolute-involute curve, Bertrand curve pair, striction curves.

AMS(2010): AMS(2010): 53A04, 53A05. §1. Introduction

A ruled surface can always be described (at least locally) as the set of points swept by a moving straight straight line. line. A ruled surface is one which can be generated generated by the motion of a straight straight line line in Euclidean 3 space [2]. [2]. Choosin Choosingg a direct directrix rix on the surfac surface, e, i.e. a smooth unit unit speed speed curve α (s) orthogonal to the straight lines and then choosing v (s) to be unit vectors along the curve in the direction of the lines, the velocity vector α s and v and v satisfy α , v = 0 wher wheree ′ αs = α . The fundamen fundamental tal forms of the B scroll with null directrix and Cartan frame in the Minkowskian 3 space are space are examined in [5]. The properties of some ruled surfaces are also 3 examined in E [6] , [7] ,[9] and [11]. A striction point on a ruled surface ϕ( ϕ (s, v ) = α( α(s) + v.e( v.e(s) is the foot of the common normal between two consecutive generators (or ruling). To illustrate the current situation, we bring here the famous example of L. K. Graves [3], so called the B scroll. scroll . The specia speciall ruled ruled surface surfacess B scroll over null curves with null rulings in 3dimension dimensional al Lorentzian Lorentzian space form has been introduced introduced by L. K. Graves. Graves. The Gauss map of B-scro B-scrolls lls has been examine examined d in [1]. Derivi Deriving ng a curve curve based based on an other curve curve is one of the main subjects in geometry. Involute-evolute curves and Bertrand curves are of these kinds. An involute of a given curve is well-known concept in Euclidean 3 space. space. We can say that evolute

−

  ′

−

−

−

−

−

1 Received Received November November

24, 2017, Accepted Accepted May 16, 2018.

34

S ¸eyda ¸ey da Kılı¸ Kıl ı¸co˘ co˘ glu, S¨ uleyman uley man S ¸enyurt ¸eny urt and Abdussame Abdu ssamett C ¸ alı¸skan skan

and involute involute are methods methods of deriving deriving a new curve based on a given curve. The involute involute of a curve is called sometimes evolvent and evolvents play a part in the construction of gears. The evolute evolute is the locus of the centers centers of osculatin osculating g circles of the given given planar curve [12]. Let α and α and ∗ α be the curves in Euclidean 3 space. space. The tangent lines to a curve α generate α generate a surface called ∗ the tangent surface of α of α.. If a curve α curve α is an involute of α of α,, then by definition α is an evolute of α∗ . Hence if we are given a curve α, then α, then its evolutes are the curves whose tangent lines intersect α orthogonally. By using a similar method we produce a new ruled surface based on an other ˜ scroll are ruled surface. The differential geometric elements of the involute D scroll are examined in [10]. It is well-known that if a curve is differentiable in an open interval at each point then a set of three mutually mutually orthogonal unit unit vectors vectors can b e constructe constructed. d. We say the set of these vectors are called Frenet Frenet frame or moving frame vectors. vectors. The rates of these frame vectors vectors along the curve define curvatures of the curve. The set whose elements are frame vectors and curvatures of a curve α is called Frenet-S Frenet-Serret erret apparatus apparatus of the curve. Let Frenet Frenet vector vector fields fields of α be V 1 (s (s) , V 2 (s (s) , V 3 (s (s) and let first and second curvatures of the curve α( α (s) be k 1 (s (s) and k2 (s (s) , respectively. respectively. Then the quantities V 1 , V 2 , V 3 , k1 , k2 are called the Frenet-Serret apparatus of the curves. If a rigid object moves along a regular curve described parametrically by α( α(s). then we know that this object has its own intrinsic coordinate system. The Frenet formulae are also well known as ˙1 V 0 k1 0 V 1 ˙2 = 0 V k1 k2 V 2

−

{

 

˙3 V

 

}

  −

  

 

−k2 0 V 3 where curvature functions are defined by k 1 (s) = V 1 (s), k 2 (s) = − 0

 

˙3 . V 2 , V

E3 and α E3 be given. If the tangent Let unit speed regular curve α : I and α ∗ : I tangent at the ∗ point α(s) to the curve α passes through the tangent at the point α (s) to the curve α ∗ then the curve α ∗ is called the involute of the curve α, for s I I provided that V 1 , V 1∗ = 0. We can write α∗ (s) = α (s) + (c ( c s)V 1 (s (s) (1.1)

→ →

→ →

∀ ∈





−

the distance between corresponding points of the involute curve in

3 E

is ([4],[8])

d α(s), α∗ (s) = c





| − s|, c = constant, = constant, ∀s ∈ I. I .

Theorem 1.1([4],[8]) The Frenet vectors of the involute α∗ , based on its evolute curve α are

    

V 1∗ = V 2 , V 2∗ =

−k1V 1 + k2V 3

V 3∗ =

k2 V 1 + k1 V 3

1

(k12 + k22 ) 2

1

(k12 + k22 ) 2

.

(1.2)

35

Tangent Vector Fields Of Striction Curves

The first and the second curvatures of involute α∗ are

k1∗ =

where (σ

′



k22

−

′



′

k1 k2

k12 + k22 k k1 k k2 , k2∗ = 2 2 1 2 = , λk1 λk1 (k (k1 + k2 ) λk1 (k (k12 + k22 )

−

(1.3)

− s)k1 > 0, 0 , k1  = 0. 0.

E3 and α∗∗ : I E3 be two C 2 class differentiable unit speed curves and Let α : I let V let V 1 (s), V 2 (s), V 3 (s) and V and V 1∗∗ (s), V 2∗∗(s), V 3 ∗∗ (s) be the Frenet frames of the curves α and α ∗∗ , respective respectively ly.. If the principal principal normal vector vector V 2 of the curve α is linearly dependent on the principal normal vector V 2∗∗ of the the curve curve α α ∗∗ , then the pair (α, ( α, α∗∗ ) is called a Bertrand curve pair [4], [8]. Also α ∗∗ is called a Bertrand mate. If the curve α ∗∗ is a Bertrand mate of α α then we may write α∗∗ (s) = α (s) + λV 2 (s (s) (1.4)

→ →

→ →

−

If the curve α curve α ∗∗ is Bertrand mate α (s) then we have

V 1∗∗ (s) , V 1 (s (s) = cos θ = constant. = constant. Theorem 1 Theorem 1..2([4],[8]) The distance between corresponding points of the Bertrand curve pair in 3 E is constant. Theorem 1.3([4]) If the second curvature k2 (s) = 0 along a curve α(s) then α(s) is called a Bertrand curve provided that nonzero real numbers λ and β λk1 + βk 2 = 1 hold along the curve α(s) where s s I . It follows that a circular helix is a Bertrand curve.



∈

Theorem 1.4([4]) Let α α : I : I E3 and α α ∗∗ : I E3 be two C 2 class differentiable unit speed curves and let the quantities V 1 , V 2 , V 3 , k1 , k2 and V 1∗∗, V 2∗∗ , V 3∗∗ , k1∗∗ , k2∗∗ be Frenet-Serret apparatus of the curves α and its Bertrand mate α ∗∗ respectively, then

→ → { {

→ → } { {

    

V 1∗∗ =

βV 1 + λV 3



λ2 + β 2

−

}

,

V 2∗∗ = V 2 , V 3∗∗ =

(1.5)

−λV 1 + βV 3 ; λk2 > 0



λ2 + β 2

The first and the second curvatures of the offset curve α∗∗ are given by

 

k1∗∗ k2∗∗

k1 λ k12 + k22 βk 1 λk2 = 2 = , (λ + β 2 ) k2 (λ2 + β 2 ) k22 1 = 2 . (λ + β 2 ) k2

−

−





(1.6)

36


Due to this theorem, we can write ∗∗

βk 1

k 1 1 − λk2 = m = m =⇒ 2∗∗ = = , k βk 1 − λk2 m 1

k2∗∗ k1∗∗

 

′

m′

=

m2 k2

− 

λ2 + β 2

⇒ dsds∗∗ = k

1

=

2

{



λ2 + β 2

}

·

A differentiable one-parameter family of (straight) lines α(u), X (u) is a correspondence that assigns to each u I a I a point α( α (u) R3 and a vector X vector X ((u) R3 , X (u) = 0, so that both α(u) and X (u) depend differentiable on u. For each each u I , the line L which passes through α(u) and is parallel to X ( X (u) is called the line of the family at u. u . Given a one-parameter family of lines α(u), X (u) the parameterized surface

∈

{

∈

∈

∈



}

ϕ(u, v ) = α( α(u) + v.X ( v.X (u) where u I and and v

∈

∈ R

(1.7)

is called the ruled surface generated by the family α(u), X (u) . The The line liness L are called the rulings and the curve α( α (u) is called an anchor of the surface ϕ, ϕ , [2].

{

}

Theorem 1 Theorem 1..5([2]) The striction point on a ruled surface ϕ( ϕ (u, v ) = α( α (u) + v.X ( v.X (u) is the foot of the common normal between two consecutive generators (or ruling). The set of striction points defines the striction curve given by c(u) = α = α((u)

′

′

u

u

αu , X u  − X .X (u) ′ , X ′ 

(1.8)

where X u′ = D = DT X (u).

angent Vector Fields of Striction Striction Curves Curves Along the Involute Involute and §2. On the Tangent Bertrandian Frenet Ruled Surfaces

Definition 2.1 In the Euclidean 3

− space, let α( α (s) be the arc length curve. The equations

 

ϕ1 (s, (s, u1 ) = α = α (s) + u1 V 1 (s (s) (2.1)

ϕ2 (s, (s, u2 ) = α = α (s) + u2 V 2 (s (s) ϕ3 (s, (s, u3 ) = α = α (s) + u3 V 3 (s (s)

−

are the parametrization of the ruled surface which is called V 1 scroll ( tangent ruled surface), V 2 scroll (normal ruled surface) and V 3 scroll (binormal ruled surface) respectively in [6].

−

−

striction curves curves of Frenet Frenet ruled ruled surface surfacess are are given given by the following following Theorem 2.1([6]) The striction

37


matrix

 

− α c2 − α c3 − α c1

 

=

 

0

0

0

0

k1 k22 +k22

0

0

0

0

  

V 1 V 2 V 3

 

.

Theorem 2.2 The tangent vector fields T 1 , T 2 and T 3 belonging to striction curves of Frenet ruled surface is given by

[T ] T ] =

 

T 1 T 2 T 3

or

where a =

k22 η c2 (s)





′

   

=

  

T 1 T 2 T 3

, b =

1

0 k1 ( η )

k22 η c2 (s)



 

1

( kη1 )

′

      

k1 k2

 c (s) ηc (s)

′

=

0

′

′

2

2

0

 

0

1

0 0

a

b

1

0 0

c

  

V 1 V 2 V 3

V 1 V 2 V 3

 

.

′

c (s) , ′

2

c =

k1 k2 η c2 (s)



′

2

2

η = k k 1 + k2 .  and η =

Proof It is easy to give this matrix matrix because we have already got the following following equalities equalities T 1 (s (s) = T 3 (s) = α′ (s) = V 1 . Since c Since c 2 (s) = α( α(s) +

c2′ (s)

k1 V 2 , where k where k 12 + 2 k1 + k22 =

T 2 (s (s) =

k22 = η = 0, hence we have



′

k22 k1 k 1 k2 V 1 + V 2 + V 3 , η η η c2′ (s) ηk22 V 1 + (k ( k1′ η k1 η ′ ) V 2 + ηk2 k1 V 3 = . 1 c2′ (s) 2 2 4 ′ 3 ′ η k2 + (k ( k1 ηk1 η )

 







−



2.1 Involute Involute Frenet Frenet Ruled Surfaces Surfaces

In this subsection, first we give the tangent, normal and binormal Frenet ruled surfaces of the involut involute-ev e-evolute olute curves. curves. Further urther we write their their parametri parametricc equations equations in terms terms of the Frenet apparatus apparatus of the involute-e involute-evolu volute te curves. Hence they are called involute Frenet ruled surfaces as in the following way.

38


Definition 2.2([6]) In the Euclidean 3-space, let α( α (s) be the arc length curve. The equations ϕ∗1 (s, (s, v1 ) = α∗ (s) + v1 V 1∗ (s) = α = α (s) + (σ (σ ϕ∗2 (s, (s, v2 ) =

− s)V 1 (s (s) + v1 V 2 (s (s) , −k1V 1 + k2V 3 α∗ (s) + v2 V 2∗ (s) = α = α (s) + (σ ( σ − s)V 1 (s (s) + v2 2 2

ϕ∗3 (s, (s, v3 ) = α∗ (s) + v3 V 3∗ (s) = α = α (s) + (σ (σ

− s)V 1 (s (s) + v3

 

1

(k1 + k2 ) 2

k2 V 1 + k1 V 3 1

(k12 + k22 ) 2

 

,

are are the para parametri metrizati zation on of the ruled ruled surface surfacess which which are are called called involute involute tangent tangent ruled ruled surface surface,, involute normal ruled surface and involute binormal ruled surface, respectively. We can deduce from Theorem 2. 2 .1 striction curves of the involute Frenet ruled surfaces are given by the following matrix

 

c∗1

− α∗ c∗2 − α∗ c∗3 − α∗

 

=

 

0

0

0

0

k1 k1 +k2 2

0

0

0

0

∗

2

∗

∗

  

V 1∗ V 2∗ V 3∗

 

.

It is easy to give the following matrix for the striction curves of four Frenet ruled surfaces along the the involute curve involute curve α α ∗ . c∗1 (s (s) = c∗3 (s) = α∗ (s) , k∗ c∗2 (s) = α∗ (s) + ∗2 1 ∗2 V 2∗ (s) . k1 + k2 Also we can write explicit equations of the striction curves on involute Frenet ruled surfaces in terms terms of Frenet apparatus apparatus of an evolute evolute curve α curve α.. Theorem 2 Theorem 2..3 The equations of the striction curves on involute Frenet ruled surfaces in terms of Frenet apparatus of an evolute curve α are given by

 

c∗1

− α c∗2 − α c∗3 − α

 

=

  

σ (σ

− s)

−

−s

k12 2 2 (1+m) (k1 +k2 )(1+m

1

σ

0

−s



0 0

0 (σ s)k1 k2 2 (k1 + k22 ) (1 + m) 0

−

   

V 1 V 2 V 3

 

.

Theorem 2 Theorem 2..4 The tangent vector fields T T 1 ∗ , T 2 ∗ , T 3 ∗ of striction curves belonging to an involute Frenet ruled surface in terms of Frenet apparatus by themselves are given by

[T ∗ ] =

 

T 1 ∗ T 2 ∗ T 3 ∗

 

=

 

1

0

0

a∗

b∗

c∗

1

0

0

  

V 1 ∗ V 2 ∗ V 3 ∗

 

.

39


′ k1 η ∗ c2 ′ (s)

   ∗

∗

a =

k2∗ 2 η ∗ c∗2 ′ (s)

 

∗

∗

, b =

k ∗ k∗ , c∗ = ∗ 1∗ ′ 2 , η ∗ = k 1∗ 2 + k2∗ 2 , µ∗ = η c2 (s)

 

2.2 Bertrandian Bertrandian Frenet Frenet ruled surfaces surfaces

k2∗ k1∗

 

′

.

In this subsection, first we give the tangent, normal and binormal Frenet ruled surfaces of the Bertrand mate α∗∗ . Further urther we write their parametric parametric equations equations in terms terms of the Frenet Frenet apparatus of the Bertrand curve α. Hence they are called Bertrandian Frenet ruled surfaces as in the following way. Definition 2 Definition 2..3([6]) In the Euclidean 3

− space, let α( α (s) be the arc length curve. The equations

∗∗ ∗∗ ϕ∗∗ = α + λV 2 + w1 1 (s, w1 ) = α (s) + w1 V 1 (s) = α

βV 1 + λV 3

 − 

λ2 + β 2 ∗∗ ∗∗ ϕ∗∗ = α + (λ (λ + w2 ) V 2 , 2 (s, w2 ) = α (s) + w2 V 2 (s) = α ∗∗ ∗∗ ϕ∗∗ = α + λV 2 + w3 3 (s, w3 ) = α (s) + w3 V 3 (s) = α

,

λV 1 + βV 3 λ2 + β 2

(2. (2.2)



,

are the parametrization of the ruled surfaces which are called Bertrandian tangent ruled surface, Bertrandian normal ruled surface and Bertrandian binormal ruled surface, respectively. We can also deduce from Theorem 2. 2 .1 the striction curves of Bertrand Frenet ruled surfaces are given by the following matrix

 

c∗∗ 1

− α∗∗ ∗∗ c∗∗ 2 −α ∗∗ c∗∗ 3 −α

 

=

 

0 0

0

0

∗∗

k1

0

k1 +k2

2

∗∗

2

∗∗

0

0 0

  

V 1∗∗ V 2∗∗ V 3∗∗

 

.

It is easy to give the following matrix for the striction curves belonging to Bertrand Frenet ruled surfaces ∗∗ ∗∗ c∗∗ (s) 1 (s) = c3 (s) = α k1∗∗ ∗∗ ∗∗ c∗∗ ( s ) = α ( s ) + 2 ∗∗2 + k ∗∗2 ∗∗2 V 2 (s) k1∗∗2 2

equations ons of the striction striction curves on Bertra Bertrandia ndian n Frenet renet ruled surfaces surfaces in Theorem 2.5 The equati terms of Frenet apparatus of curve α

 

c∗∗ 1

−α c∗∗ 2 −α c∗∗ 3 −α

 

    0

=

0 0

λ λ+

m(λ2 +β 2 )k2 (m2 +1)

λ

     0

V 1

0

V 2

0

V 3

 

.

40


Proof Since Since the equations equations of the striction striction curves on Bertrandian Bertrandian Frenet Frenet ruled surfaces in terms of Frenet apparatus of curve α are ∗∗ ∗∗ c∗∗ (s) = α (s) + λV 2 (s (s) 1 (s) = c3 (s) = α

the first and the second curvatures of the curve α∗∗ are given by k1∗∗ = 1 1 ∗∗ . Also k Also k k = and 2 2 (λ2 + β 2 ) k2 (λ2 + β 2 )

k2∗∗ =

∗∗ c∗∗ 2 (s) = α (s) +

k1∗∗ ∗∗ ∗∗2 + k∗∗2 ∗∗2 V 2 (s) = α k1∗∗2 2

+

      −  

T 1∗∗ T 2∗∗ T 3∗∗

 

=

 

2

(βk 1

vector fields T 1∗∗, T 2∗∗ and Theorem 2.6 The tangent vector to Bertrandian Frenet ruled surface are given by

 

λ2 + β 2 k2

λ+

1

0

0

a∗∗

b∗∗

c∗∗

1

0

0

  

−

βk 1 λk2 and (λ2 + β 2 ) k2

λk2 ) + 1

V 2 .

striction curves belonging belonging T 3∗∗ of striction

V 1∗∗ V 2∗∗ V 3∗∗

 

where ′ k1 η ∗∗ c2 ′ (s)

    ∗∗

a∗∗ =

k2∗∗ 2 ′ η∗∗ c∗∗ 2 (s)

 

, b∗∗ =

∗∗

, c∗∗ =

k1∗∗ k2∗∗ ′ η∗∗ c∗∗ 2 (s)

 

and η∗∗ = k 1∗∗ 2 + k2∗∗ 2 .

product of tangent tangent vector vector fields fields T 1 ∗ , T 2 ∗ , T 3∗ and tangent tangent vector vector fields fields Theorem 2.7 The product ∗∗ ∗∗ ∗∗ striction curves on an involute involute and Bertrandia Bertrandian n Frenet renet ruled ruled surface surface reT 1 , T 2 , T 3 of striction spectively, are given by

T

[T ∗ ] [T ∗∗ ] = A

 

0

Ab∗∗

0

B

a∗∗ B + b∗∗ a∗ A + c∗∗ C

B

0

b∗∗ A

0

 

where the coefficients are A =



(λ2 + β 2 )(k )(k1 2 + k2 2 ) , B = b ∗ ( βk 1 + λk2 ) + c∗ , C = = b ∗ + c∗ ( λk2 + βk 1 ).

−

−

Proof Let [T [T ∗] = [A∗ ] [V ∗ ] and [T [T ∗∗] = [A∗∗ ] [V ∗∗ ] be given. By using the properties of a matrix following result can be obtained:

41


[T ∗ ] [T ∗∗ ]T

= [A∗ ] [V ∗ ] ([A ([A∗∗ ] [V ∗∗ ])T



       T

= [A∗ ] [V ∗ ] [V ∗∗ ]

=

=

   

= A

1

0

0

a∗

b∗

c∗

1

0

0

1

0

0

a∗

b∗

c∗

1

0

0

 

T

[A∗∗ ] V 1 ∗ V 2 ∗ V 3 ∗ V 1

∗

V 2 ∗ V 3 ∗

b∗∗ A

0

      

  

V 1 ∗∗

1

0

0

a∗∗

b∗∗

c∗∗

1

0

0

1

0

0

a∗∗

b∗∗

c∗∗

1

0

0

V 1

∗∗

V 2 ∗∗ V 3 ∗∗

T

    

0

B

a∗∗ B + b∗∗ a∗ A + c∗∗ C

B

0

b∗∗ A

0

V 2 ∗∗ V 3 ∗∗

   

T

T

.

As a result of Theorem 2. 2 .1 we can write that in the Euclidean 3 space, space, the position of ∗ ∗ ∗ ∗∗ ∗∗ ∗∗ the unit tangent vector field T 1 , T 2 , T 3 and T and T 1 , T 2 , T 3 of striction curves belonging to ruled ∗∗ ∗∗ surfaces ϕ surfaces ϕ ∗1 , ϕ∗2 , ϕ∗3 and ϕ and ϕ ∗∗ respectively, along the curve α curve α ∗ and α and α ∗∗ , can be expressed 1 , ϕ2 , ϕ3 respectively, by the following equations

−

[T ∗] [T ∗∗ ]T =

   

T 1∗ , T 1∗∗

 T 1∗, T 2∗∗ T 1∗, T 3∗∗ T 2∗ , T 1∗∗ T 2∗, T 2∗∗  T 2∗, T 3∗∗  T 3∗ , T 1∗∗ T 3∗, T 2∗∗  T 3∗, T 3∗∗ 

 

,

T

here [T [T ∗∗] is the transpose matrix of [T [ T ∗∗ ] . Hence we may write that, there are four tangent vector fields on striction curves which are perpendicular to each other, for the involute and Bertrandian Frenet ruled surfaces given above. Since T 1∗ , T 1∗∗ = T 1∗ , T 3∗∗ = T 3∗ , T 1∗∗ = T 3∗ , T 3∗∗ = 0, it is trivial.



 

 

 



tangent vector vector fields of striction striction curves on an involute involute tangent tangent and Theorem 2.8 (i) The tangent Bertrandian normal ruled surfaces are perpendicular under the condition



(βk 1 − λk2 )(λ )(λ2 + β 2 )k2 (βk 1 − λk2 )2 + 1



′

= 0, 0 , λ2 =

−β 2

or k1 2 =

−k22.

(ii) ii) The tangent vector fields of striction curves on an involute binormal and Bertrandian normal ruled surfaces are perpendicular under the condition



(βk 1 λk2 )(λ )(λ2 + β 2 )k2 (βk 1 λk2 )2 + 1

−

−



′

= 0, 0 , λ2 =

−β 2

or k1 2 =

−k22.

42


Proof (i) Since T 1∗ , T 2∗∗ = b = b ∗∗ A and T 1∗, T 2∗∗ = 0



 





(βk 1 λk2 )(λ )(λ2 + β 2 )k2 (βk 1 λk2 )2 + 1

−

−

(βk 1 λk2 )(λ )(λ2 + β 2 )k2 (βk 1 λk2 )2 + 1

−

−





b∗∗ A = 0

′

 

(λ2 + β 2 )(k )(k1 2 + k2 2 ) = 0

′

= 0 or



(λ2 + β 2 )(k )(k1 2 + k2 2 ) = 0,

this completes the proof. (ii) ii) Since T 1∗ , T 2∗∗ = T 3∗, T 2∗∗ = b = b ∗∗ A, the proof is trivial.



 



tangent vector vector fields of striction striction curves curves on an involute involute normal and Theorem 2.9 (i) The tangent Bertrandian tangent ruled surfaces are perpendicular under the condition 3

−βk1 + λk2 =

k22 ( kk12 )′ (k12 + k22 ) 2



(k12 + k22 )3 + k24 ( kk12 )′

2

 

2 1

2 2

λk1 (k +k )

k

2 1

′.

5 2

(k +k22 )3 +k24 ( k1 )

′

2

2





(ii) ii) The tangent vector fields of striction curves on an involute normal and Bertrandian binormal ruled surfaces are perpendicular under the condition 3

−βk1 + λk2 =

k22 ( kk12 )′ (k12 + k22 ) 2



(k12 + k22 )3 + k24 ( kk12 )′

2

 

2 1

2 2

λk1 (k +k )

k

2 1

′.

5 2

(k +k22 )3 +k24 ( k1 )

′

2

2





Proof (i) Since T 2∗ , T 1∗∗ = B = B = = b ∗ ( βk 1 + λk2 ) + c∗ and T 2∗ , T 1∗∗ = 0





−





B = b = b ∗ ( βk 1 + λk2 ) + c∗

−

βk 1

− λk2 +

k22 ( kk12 )′ (k12 + k22 )



(k12 + k22 )3 + k24 ( kk12 )′

−βk1 + λk2 =

2

 

2 1

2 2

λk1 (k +k )

k

(k12 + k22 )3 + k24 ( kk12 )′

this completes the proof.

2

′

2

3

 

2



λk1 (k12 +k22 ) 2 1



5 2

(k12 +k22 )3 +k42 ( k1 )

k22 ( kk12 )′ (k12 + k22 ) 2



= 0

3 2

= 0

′

′,

5 2

k

(k +k22 )3 +k24 ( k1 ) 2

′

2





(ii) ii) Since T 2∗ , T 1∗∗ = T 2∗, T 3∗∗ = B = B = = b b ∗ ( βk 1 + λk2 ) + c∗ , the proof is trivial.



 



−

product between between tangent vector fields of striction curves on an involute Corollary 2 Corollary 2..1 The inner product


43

normal and Bertrandian normal ruled surfaces of the (α∗ , α∗∗ ) is

T 2∗, T 2∗∗ = a = a∗∗ B + b∗∗ a∗ A + c∗∗ C. References [1] Alias L.J., Ferrandez A., Lucas P. and Merono M. A., On the Gauss map of B-scrolls, Tsukuba J. Math. , 22, 371-377, 1998. [2] Do Carmo, M. P., Differential Geometry of Curves and Surfaces , Prentice-Hall, ISBN 013-212589-7, 1976. [3] Graves L.K., L.K., Codimension one isometric immersions between between Lorenzt spaces, Trans. Amer. Math. Soc., 252, 367-392, 1979. ¨ [4] Hacısaliho Hacısaliho˘˘glu glu H.H., Diferensiyel Geometri , Cilt 1, Ín¨ on¨ onü Universitesi Yayinlari, Malatya, 1994. [5] Kılı¸ Kılı¸co˘ co˘glu glu S¸, ¸, Hac Hacısal ısaliho˘ iho˘glu H.H. and S¸enyurt S., On the fundamental forms of the B-scroll with null directrix and Cartan frame in Minkowskian 3-space, Applied Mathematical Sciences , doi.org/10.12988/ams.2015.53230, 9(80), 3957 - 3965, 2015. [6] Kılı¸ Kılı¸co˘ co˘glu glu S ¸, ¸, S ¸enyurt, ¸e nyurt, S. and Hacısaliho˘ Hacısali ho˘glu glu H.H., On the striction curves of Involute and Mathematical Sciences Sciences , 9(142), 7081 Bertrandian Frenet ruled surfaces in E3 , Applied Mathematical 7094, 2015, http://dx.doi.org/10.12988/ams.2015.59606. [7] Kılı¸ Kılı¸co˘ co˘glu glu S ¸ , S¸ enyurt, enyur t, S. S . and a nd C ¸ alı¸skan, skan, A., On the strictio stri ction n curve c urvess of Involutive Involu tive Frenet ruled r uled 3 surfaces in E , Konuralp Journal of Mathematics , 4(2), 282-289, 2016. [8] Lipschutz Lipschutz M.M., Differential Geometry , Schaum’s Outlines. [9] S ¸enyurt, ¸enyurt, S. and C ¸ alı¸skan, skan, A., A new approach on the striction curves along Bertrandian Darboux Frenet ruled surface, —it AIP Conference Proceedings, 1726, 020032 (2016), doi: 10.1063/1.4945858. ˜ scroll, [10] S ¸enyurt, ¸enyu rt, S. and Kılı¸co˘glu glu S¸., ¸., On the differential geometric g eometric elements ele ments of the involute D scroll , Adv. Appl. Cliff ord Algebras , 2015 Springer Basel,doi:10.1007/s00006-015-0535-z. [11] S ¸enyurt, ¸enyurt, S., On involute B-scroll a New View, University University of Ordu, Journal of Science and Technology , 4(1), 59-74, 2014. [12] Springerlink, Encyclopaedia of Mathematics , Springer-Verlag, Springer-Verlag, Berlin Heidelberg, New York, 2002.


On the Leap Zagreb Indices of Generalized xyz -Point-Line -Point-Line Transformation ransformation Graphs T xyz (G) when z = = 1 B. Basavanagoud and Chitra E. (Department of Mathematics, Karnatak University, Dharwad - 580 003, Karnataka, India) E-mail: [email protected], [email protected]

Abstract: Abstract: For a graph G, the first, second and third leap Zagreb indices are the sum of squares of 2-distance degree of vertices of G; G ; the sum of product of 2-distance degree of end vertices of edges in G and the sum of product of 1-distance degree and 2-distance degrees of vertices of G, respectiv respectively ely.. In this paper, we obtain the expressio expressions ns for these these three leap Zagreb indices of generalized xyz point xyz point line transformation graphs T xyz (G) when z when z = = 1.

Key Words: Words: Distance, degree, diameter, Zagreb index, leap Zagreb index, reformulated Zagreb index.

AMS(2010): AMS(2010): 05C90, 05C35, 05C12, 05C07. §1. Introduction

Let G = (V, E ) be a simple graph of order n and size m. The k-distance degree of a vertex v V ( V (G), denoted by dk (v/G) v/G) = N k (v/G) v/G) where N k (v/G) v/G) = u V ( V (G) : d( d (u, v ) = k [17] in which d(u, v ) is the distance between the vertices u and v in G that is the length of the shortest path joining u and v in G. The The degree of a vertex v in a graph G is the number of edges incident to it in G and is denoted by dG (v). Here Here N 1 (v/G) v/G) is nothing but N G (v) and d1 (v/G) v/G) is same as d G (v). If u u and v are two adjacent vertices of G of G,, then the edge connecting them will be denoted by uv. uv . The degree of an edge e = uv = uv in in G G,, denoted by d by d 1 (e/G) e/G) (or d (or d G (e)), is defined by d 1 (e/G) e/G) = d 1 (u/G u/G)) + d1 (v/G) v/G) 2. The complement of a graph G graph G is is denoted by G by G whose whose vertex set is V ( V (G) and two vertices of G are adjacent if and only if they are nonadjacent in G. G. G has n has n vertices vertices and n(n2−1) m edges. The line graph L(G) of a graph G graph G with with vertex set as the edge set of G of G and and two vertices of L of L((G) are adjacent whenever the corresponding edges in G have have a vertex vertex incident incident in comm common. on. The complement of line graph L(G) or jump graph J (G) of a graph G graph G is a graph with vertex set as the edge set of G and G and two vertices of J of J ((G) are adjacent whenever the corresponding edges in G have no vertex incident in common. The subdivision graph S (G) of a graph G graph G whose whose vertex set is V is V ((G) E (G) where two vertices are adjacent if and only if one is a vertex of G of G and other is

∈

|

|

{ ∈

}

−

−



1 Support Supported ed

by the Unive Universit rsity y Grants Grants Commis Commission sion (UGC), (UGC), New Delhi, Delhi, throug through h UGC-SA UGC-SAP P DRS-III DRS-III for 2016-2021: 2016-2021: F.510/3/DRS-III F.510/3/DRS-III/2016 /2016(SAP(SAP-I) I) and the DST INSPIRE Fellowsh Fellowship ip 2017: No.DST/INS No.DST/INSPIRE PIRE FelFellowship/[IF170465]. 2 Received February 16, 2017, Accepted May 18, 2018.

On the Leap Zagreb Indices of Generalized xy z-Point-Line -Point-Line Transformation Transformation Graphs T T xyz (G) when z when z = 1

45

an edge of G G incident with it. The partial complement of subdivision graph S (G) of a graph G graph G whose vertex set is V ( V (G) E (G) where two vertices are adjacent if and only if one is a vertex of G G and the other is an edge of G G non incident with it. We follow [11] and [13] for unexplained graph theoretic terminologies and notations. The first and second Zagreb indices [9] of a graph G are defined as follows:





M 1 (G) =

dG (v)2

and M 2 (G) =

( G) v∈V (G



dG (u)dG (v),

uv∈ uv∈E (G)

respectively. These are widely studied degree based topological indices due to their applications in chemistry. For details see the papers [5, 7, 8, 10, 18]. The first Zagreb index [15] can also be expressed as



M 1 (G) =

[dG (u) + dG (v)]

uv∈ uv∈E (G)

Ashrafi et al. [1] defined the first and second Zagreb coindices as M 1 (G) =





[dG (u) + dG (v)] and M 2 (G) =

uv uv ∈E (G)

[dG (u)dG (v)],

uv uv ∈E (G)

respectively. In 2004, Milićevi´ c et al. [14 [14]] reform reformula ulated ted the Zag Zagreb reb indices indices in terms terms of edge-d edge-degr egrees ees instead of vertex-degrees. The first and second reformulated Zagreb indices are defined, respectively, as EM 1 (G) =



dG (e)2

and EM 2 (G) =



[dG (e)dG (f )] f )]

e∼f

e∈E (G)

In [12], Hosamani and Trinajstić defined the first and second reformulated Zagreb coindices respectively as EM 1 (G) =

 

[dG (e) + dG (f )] f )],,

e∼f

EM 2 (G) =

[dG (e) + dG (f )] f )]..

e≁f

In 2017, Naji et al. [16 [16]] introdu introduced ced the leap leap Zagreb indic indices. es. For a graph graph G, the first, second, and third leap Zagreb indices [16] are denoted and defined respectively as: LM 1 (G) =

  

d2 (v/G) v/G)2 ,

( G) v∈V (G

LM 2 (G) =

d2 (u/G u/G))d2 (v/G) v/G),

uv∈ uv∈E (G)

LM 3 (G) =

d1 (v/G) v/G)d2 (v/G) v/G).

( G) v∈V (G

Throughout this paper, in our results we write the notations d 1 (v) and d1 (e) respectively for degree of a vertex v and degree of an edge e of a graph.

46

B. Basavanagoud Basavanagoud and Chitra E.

§2. Generalized xyz-Point-Line xyz -Point-Line Transformation Graph T xyz (G)

The procedure of obtaining a new graph from a given graph by using incidence (or nonincidence) relation between vertex and an edge and an adjacency (or nonadjacency) relation between two vertices or two edges of a graph is known as graph transformation and the graph obtained by doing so is called a transformation transformation graph. graph. For a graph G = (V, E ), ), let G0 be the graph with V ( V (G0 ) = V ( V (G) and with no edges, G edges, G 1 the complete graph with V with V ((G1 ) = V ( V (G), G ), G + = G, G, and − G = G. G. Let denotes the set of simple graphs. The following graph operations depending on x,y,z 0 , 1, + , induce functions T functions T xyz : . These operations are introduced by Deng et al. in [6]. They called called these these resultin resulting g graphs as xyz as xyz-trans -transform formation ationss of G of G, denoted by T by T xyz (G) = Gxyz and studied the the Laplacian characteristic polynomials and some some other Laplacian Laplacian parameters of xyz xy z -transformations of an r-regular r -regular graph G graph G.. In [2], Wu Bayoindu Bayoindureng reng et al. introduced introduced the total transformation graphs and studied the basic properties of total transformation graphs. Motiva Motivated ted by this, this, Basavanagou Basavanagoud d [3] studied studied the basic properties properties of the xyz-transformation xyz -transformation graphs by calling them xyz-point-line xyz -point-line transformation graphs by changing the notion of xyzxyz xyz transformations of a graph G as T (G) to avoid confusion between parent graph G and its xyz-transformations. xyz -transformations.

∈ {

G

−}

G →G

Definition 2 Definition 2..1([6]) Given a graph G G with vertex set V ( V (G) and edge set E E (G) and three variables xyz x,y,z 0, 1, +, , the xyz-point-line transformation graph T (G) of G is the graph with vertex set V ( V (T xyz (G)) = V ( V (G) E (G) and the edge set E ( E (T xyz (G)) = E ((G ((G)x ) E ((L ((L(G))y ) E (W ) W ) where W = S (G) if z = +, W = S (G) if z = , W is the graph with V ( V (W ) W ) = V ( V (G) E (G) and with no edges if z = 0 and W is the complete bipartite graph with parts V ( V (G) and E ( E (G) if z z = 1.

∈ {

−}

∪

∪

−

∪

∪

Since there are 64 distinct 3 - permutations of 0, 1, +, . Thus Thus obtain obtained ed 64 kinds kinds of generalized xyz-poin xyz -point-li t-line ne transformation transformation graphs. graphs. There are 16 different different graphs for each case when z when z = 0, z 0, z = 1, z 1, z = +, z +, z = .

{

−}

−

In this paper, we consider the xyz xy z -point-line transformation graphs T xyz (G) when z when z = 1. Example 2.1 Let G = K 2 K 3 be a graph. Then Then G0 be the graph with V ( V (G0 ) = V ( V (G) and 1 1 + − with no edges, G the complete graph with V ( V (G ) = V ( V (G), G = G, G , and G and G = G which are depicted in the following Figure 1.

·

Figure 1

x yz-point-line oint-line transformation transformation graphs graphs The self-explanatory examples of the path P 4 and its xyz-p (P 4 ) are depicted in Figure 2.

xy1 xy1

T


Figure 2

47

48


xy 1 §3. Leap Zagreb Zagreb Indices of T xy1 (G)

Theorem 3.1([3]) Let G G be a graph of order n and size m m . Then (1) V ( V (T xyz (G)) = n = n + m;

| | (2) |E (T xyz (G))| = |E (Gx )| + |E (L(G)y )| + |E (W ) W )|, where

    |     −     |   −  −   |   − 0

x

|E (G ) =

n 2

m

n 2

if x = 0. if x = 1.

if x = +.

m if x =

0

|E (L(G)y ) =

|E (W ) W ) =

if y = 0.

m 2

if y = 1.

m + 21 M 1

m+1 2

−.

if y = +.

1 2 M 1

if y =

0

if z = 0.

mn

if z = 1.

m

if z = +.

m(n

2) if z =

−.

−.

xy 1 The following Propositions are useful for calculating d 2 (T xy1 (G)) in Observation 3. 3 .4.

Proposition 3 Proposition 3..2([4]) Let G be a graph of order n and size m m . Let v be a vertex of G. Then

dT xy1 (G) (v) =

   

m

  

n

n+m

if x = 0, y

−1

m + dG (v) n+m

− 1 − dG(v)

∈ {0, 1, +, −} if x = 1, y ∈ {0, 1, +, −} if x = +, y ∈ {0, 1, +, −} if x = −, y ∈ {0, 1, +, −}

Proposition 3 Proposition 3..3([4]) Let G be a graph of order n and size m m . Let e be an edge of G . Then

dT xy1 (G) (e) =

n+m

−1

n + dG (e) n+m

∈ {0, 1, +, −} if y = 1, x ∈ { 0, 1, +, −} if y = +, x ∈ {0, 1, +, −} if y = −, x ∈ {0, 1, +, −} if y = 0, x

− 1 − dG(e)

Observation 3.4 Let Let G be a connected (n, (n, m) graph. Then

 

(1) d2 (v/T 001)(G )(G)=

(n

− 1) (m − 1)

if v

∈ V ( V (G) if v = e = e ∈ E (G)


 ∈  − ∈  −− ∈  − ∈  ∈  − − ∈ ∈  ∈  ∈  ∈  −− ∈  ∈  ∈ ∈  − ∈  − − ∈  ∈  − − ∈  −− ∈  − − ∈  ∈ ∈  −− − ∈  ∈  ∈  ∈  −− ∈  ∈  ∈  ∈

(2) d2 (v/T 101)(G )(G)=

(3) d2 (v/T +01)(G )(G)=

(4) d2 (v/T −01 )(G )(G)= (5) d2 (v/T 011)(G )(G)= (6) d2 (v/T 111)(G )(G)=

(7) d2 (v/T +11)(G )(G)=

(8) d2 (v/T −11 )(G )(G)= (9) d2 (v/T 0+1)(G )(G)=

(10) d2 (v/T 1+1 )(G )(G)=

(11) d2 (v/T ++1)(G )(G)= (12) d2 (v/T −+1 )(G )(G)= (13) d2 (v/T 0−1 )(G )(G)= (14) d2 (v/T 1−1 )(G )(G)=

(15) d2 (v/T +−1 )(G )(G)= (16) d2 (v/T +−1 )(G )(G)=

0

if v

(m

V ( V (G)

1) if v = e = e

n

1

(m

d1 (v/G) v/G) if v

1)

1

E (G)

V ( V (G)

0 if v = e = e n

E (G)

V ( V (G)

if v = e = e

0 if v

E (G)

d1 (v/G) v/G) if v

V ( V (G)

0

if v = e = e

d1 (v/G) v/G) if v

V ( V (G)

0

if v = e = e

n

1

m

1

E (G)

V ( V (G)

if v = e = e

1 if v

0

V ( V (G)

if v = e = e

d1 (v/G) v/G) if v

n

E (G)

1)

(m

E (G)

E (G)

if v

V ( V (G)

d1 (e/G) e/G) if v = e = e

0

if v

E (G)

V ( V (G)

m

1

d1 (e/G) e/G) if v = e = e

n

1

d1 (v/G) v/G)

m

1

d1 (e/G) e/G) if v = e = e

if v

d1 (v/G) v/G)

m

n

1

1

0

if v

E (G)

V ( V (G)

E (G)

d1 (v/G) v/G) if v

V ( V (G)

d1 (e/G) e/G)

if v = e = e

d1 (v/G) v/G) if v

V ( V (G)

d1 (e/G) e/G)

E (G)

V ( V (G)

d1 (e/G) e/G) if v = e = e 1

E (G)

V ( V (G)

d1 (e/G) e/G) if v = e = e if v

E (G)

V ( V (G)

if v

d1 (e/G) e/G) if v = e = e

n

49

if v = e = e

E (G)

E (G)

The above Observation 3. 3 .4 is useful for computing leap Zagreb indices of transformation xy1 xy1 graphs T graphs T (G) in the forthcoming theorems.

graph. Then Theorem 3.5 Let G be (n, ( n, m) graph.

50


(1) LM (1) LM 1 (T 001 (G)) = n( n (n

− 1)2 + m(m − 1)2; (2) LM (2) LM 2 (T 001 (G)) = mn( mn (m − 1)(n 1)(n − 1); (3) LM (3) LM 3 (T 001 (G)) = mn( mn (m + n − 2). 2). Proof The graph T 001 (G) has n + m vertices and mn edges, edges, refer refer Theore Theorem m 3 .1. By definitions of the first, second and the third leap Zagreb indices along with Propositions 3 .2, 3.3 and Observation 3. 3 .4 we get the following.

 

LM 1 (T 001 (G)) =

d2 (v/T 001 (G))2

v∈V ( V (T 001 (G))

d2 (v/T 001 (G))2 +

=

v∈V ( V (G)

= n(n

LM 2 (T 001 (G)) =

     



d2 (e/T 001(G))2

e∈E (G) 2

− 1)

+ m(m

2

− 1) .

d2 (u/T 001(G)) d2 (v/T 001(G))



uv∈ uv∈E (T 001 (G))



d2 (u/T 001(G)) d2 (v/T 001 (G))

=

 

uv∈ uv∈E (S (G))



d2 (u/T 001(G)) d2 (v/T 001(G))

+


= (n



− 1)(m 1)(m − 1)2m 1)2m + (n (n − 1)(m 1)(m − 1)(mn 1)(mn − 2m) = mn = mn((n − 1)(m 1)(m − 1). 1).

LM 3 (T 001 (G)) =

   

d1 (v/T 001 (G)) d2 (v/T 001 (G))

v∈V (T (T 001(G))

=





d1 (v/T 001 (G)) d2 (v/T 001 (G))

(G) v∈V (G

 



d1 (e/T 001(G)) d2 (e/T 001(G))

+

e∈E (G)

= mn( mn(n



− 1) + mn( mn(m − 1) = mn( mn(m + n − 2). 2).

graph. Then Theorem 3.6 Let G be (n, ( n, m) graph. (1) LM (1) LM 1 (T 101 (G)) = m( m (m

− 1)2;

(2) LM (2) LM 2 (T 101 (G)) = 0; (3) LM (3) LM 3 (T 101 (G)) = mn( mn (m

− 1). 1).

Proof Notice Notice that that the graph graph T 101 (G) has n + m vertices vertices and mn + mn + n(n2−1) edges by Theorem 3. 3.1. According According to the definitions definitions of first, second and third leap Zagreb indices indices along


51

with Propositions Propositions 3.2, 3.3 and Observation 3. 3 .4, calculation shows the following.

 

LM 1 (T 101 (G)) =

d2 (v/T 101 (G))2

v∈V ( V (T 101 (G))

d2 (v/T 101 (G))2 +

=

v∈V ( V (G)

= m(m

LM 2(T 101 (G)) =

d2 (e/T 101(G))2

e∈E (G) 2

− 1) .

       

d2 (u/T 101(G)) d2 (v/T 101(G))

uv∈ uv∈E (T 101 (G))





d2 (u/T 101 (G)) d2 (v/T 101 (G))

=

 

uv∈ uv∈E (G)

+

d2 (u/T 101(G))

uv∈ / E (G)

+ +



d2 (v/T 101 (G))



d2 (u/T 101(G)) d2 (v/T 101(G))


 

 

d2 (u/T 101(G)) d2 (v/T 101(G)) = 0.


LM 3 (T 101 (G)) =



   

d1 (v/T 101(G)) d2 (v/T 101(G))

v∈V ( V (T 101 (G))

d1 (v/T 101(G))

=

v∈V ( V (G)





d2 (v/T 101(G))

 



d1 (e/T 101(G)) d2 (e/T 101(G)) = mn = mn((m

+

e∈E (G)



− 1). 1).


(1) LM (1) LM 1 (T +01 (G)) = n( n (n

− 1)2 + m(m − 1)2 + M 1(G) − 4m(n − 1);

(2) LM (2) LM 2 (T +01 (G)) = M 2(G)

− (n − 1)M 1)M 1 (G) + m[(n [(n − 1)2 + (m (m − 1)(n 1)(n2 − n − 2m)];

(3) LM (3) LM 3 (T +01 (G)) = m[ m [n(n + m)

− 2(m 2(m + 1)] − M 1 (G).

Proof By Theorem 3. 3.1, we know that the graph T +01(G) has n has n + m vertices and m and m((n + 1) edges. edges. By using using the definitio definitions ns of first, first, second second and third third leap leap Zagreb Zagreb indices indices and applying applying

52


Propositions 3. 3.2, 3.3 and Observation 3. 3 .4 we get the following.

   −−  −

LM 1(T +01 (G)) =

d2 (v/T +01 (G))2

v∈V ( V (T +01(G))

d2 (v/T +01 (G))2 +

=

v∈V ( V (G)

=

(n

1

e∈E (G)

d1 (v/G)) v/G)) +

v∈V ( V (G)

= n(n

LM 2 (T +01(G)) =

− 1)

(m

e∈E (G)

1)2 + d1 (v/G) v/G)2

(n 2

d2 (e/T +01(G))2

2

v∈V ( V (G)

=

 

+ m(m

− 1)

2

− 1)2

− 2(n 2(n − 1)d 1)d1 (v/G) v/G)

+ M 1 (G)

− 4m(n − 1). 1).

          − − −  − −−  − −−

  +

(m

e∈E (G)

− 1)2

d2 (u/T +01 (G)) d2 (v/T +01(G))



uv∈ uv∈E (T +01(G))

d2 (u/T +01(G)) d2 (v/T +01 (G))

=



uv∈ uv∈E (G)

+

d2 (u/T +01(G)) d2 (v/T +01 (G))


+

[d2 (u/T +01(G))][d ))][d2 (v/T +01 (G))]




=

1)2

(n

(n

uv∈ uv∈E (G)

+

(m



·

1)(d 1)(d1 (u/G u/G)) + d1 (v/G)) v/G)) + d1 (u/G u/G)) d1 (v/G) v/G)

1)(n 1)(n

1

d1 (u/( u/(G))

1)(n 1)(n

1

d1 (u/( u/(G))


+

(m


= M 2 (G)

LM 3 (T +01 (G)) =

− (n − 1)M 1)M 1 (G) + m[(n [(n − 1)2 + (m (m − 1)(n 1)(n2 − n − 2m)]. )].

    

d1 (v/T +01(G)) d2 (v/T +01 (G))

v∈V ( V (T +01 (G))

=



d1 (v/T +01(G)) d2 (v/T +01(G))

v∈V ( V (G)

 

d1 (e/T +01(G))

+

e∈E (G)

=



[(m [(m + d1 (v/G))( v/G))(n n

v∈V ( V (G)

= m[n(n + m)

(1) LM (1) LM 1 (T −01 (G)) = M 1 (G) + m(m

d2 (e/T +01(G))

− 1)2;



− 1 − d1(v/G))] v/G))] +

− 2(m 2(m + 1)] − M 1 (G).






e∈E (G)

n(m

− 1)


53

(2) LM (2) LM 2 (T −01 (G)) = M 2 (G) + 2m 2 m2 (m

− 1); (3) LM (3) LM 3 (T −01 (G)) = m[ m[n(m + 1) + 2(m 2( m − 1)] − M 1 (G). Proof We know the graph T −01 (G) has n + m + m vertices and (n ( n 1)( n2 + m) m) edges, refer Theorem 3. 3.1. By definitions definitions of the first, second and third leap Zagreb indice indicess and applying applying Propositions 3. 3.2, 3.3 and Observation 3. 3 .4 we have the following.

−

 

LM 1(T −01 (G)) =

v∈V ( V (T

d2 (v/T −01 (G))2

01

−

(G))

d2 (v/T −01 (G))2 +

=

v∈V ( V (G)

d2 (e/T −01 (G))2

e∈E (G)

= M 1(G) + m(m

LM 2 (T −01 (G)) =



2

− 1) .

                 −  − d2 (u/T −01 (G)) d2 (v/T −01 (G))

01

uv∈ uv∈E (T

−

(G))

d2 (u/T −01(G)) d2 (v/T −01 (G))

=

uv∈ / E (G)

+

d2 (u/T −01(G)) d2 (v/T −01 (G))


+

d2 (u/T −01(G)) d2 (v/T −01 (G))


=

[d1 (u/G u/G)] )] [d1 (v/G)] v/G)] +

uv∈ / E (G)

+

(m

1)d 1)d1 (u/G u/G))


(m

1)d 1)d1 (u/G u/G))


= M 2 (G) + 2m 2 m2 (m

LM 3(T −01 (G)) =

   

v∈V ( V (T

01

−

− 1). 1).

d1 (v/T −01 (G)) d2 (v/T −01 (G))

(G))





d1 (v/T −01 (G)) d2 (v/T −01 (G))

=

v∈V ( V (G)

 



d1 (e/T −01(G)) d2 (e/T −01 (G))

+

e∈E (G)

= m[n(m + 1) + 2(m 2( m

− 1)] − M 1(G).



graph. Then Theorem 3.9 Let G be (n, ( n, m) graph. (1) LM (1) LM 1 (T 011 (G)) = n( n (n (2) LM (2) LM 2 (T 011 (G)) = 0;

− 1)2;

(3) LM (3) LM 3 (T 011 (G)) = mn( mn (n

− 1). 1).

Proof We are easily easily know that the graph T 011 (G) has n + m + m vertices and m( m2−1 + n) n )

54


edges by Theorem 3. 3 .1. By definitio definitions ns of the first, first, second second and the third third leap Zagreb indic indices es along with Propositions Propositions 3.2, 3.3 and Observation 3. 3 .4 we know the following.

 

LM 1 (T 011 (G)) =

d2 (v/T 011 (G))2

v∈V ( V (T 011 (G))

d2 (v/T 011 (G))2 +

=

v∈V ( V (G)

= n(n

LM 2(T 011 (G)) =

− 1)2.

         

d2 (u/T 011(G)) d2 (v/T 011(G))





d2 (u/T 011(G)) d2 (v/T 011 (G))

=

   

uv∈ uv∈E (L(G))

uv∈ / E (L(G))


d2 (u/T 011(G)) d2 (v/T 011(G)) = 0.

+


   

d1 (v/T 011 (G)) d2 (v/T 011 (G))

v∈V (T ( T 011 (G))





d1 (v/T 011 (G)) d2 (v/T 011 (G))

 



d1 (e/T 011(G)) d2 (e/T 011 (G)) = mn = mn((n

+

e∈E (G)

  

d2 (u/T 011(G)) d2 (v/T 011(G))

+

v∈V (G ( G)



d2 (u/T 011(G)) d2 (v/T 011(G))

+

=

d2 (e/T 011(G))2

e∈E (G)

uv∈ uv∈E (T 011 (G))

LM 3 (T 011(G)) =





− 1). 1).

Theorem 3.10 Let G be (n, m) graph. Then LM 1 (T 111 (G)) = LM 2 (T 111 (G)) = LM 3(T 111 (G)) = 0. 0.

Proof Notice that the graph T 111 (G) has n has n + m vertices and n(n2−1) + m(m2−1) + mn edges mn edges by Theorem 3. 3.1. By definitions definitions of the first, second and third leap Zagreb indices indices along with Propositions 3. 3.2, 3.3 and Observation 3. 3 .4, we get similarly the desired result as the proof of above theorems. Theorem 3.11 Let G be (n, m) graph. Then (1) LM (1) LM 1 (T +11 (G)) = (n (n 1)(n 1)(n2 n 4m) + M 1 (G); (2) LM (2) LM 2 (T +11 (G)) = m( m (n 1)2 (n 1)M 1)M 1 (G) + M 2 (G); +11 (3) LM (3) LM 3 (T (G)) = m[( m [(n n 1)(n 1)(n + 2) 2m] M 1(G).

−

− − − − − − −

−

Proof Clearly, the graph T +11(G) has n has n + m vertices and

m(m+1) 2

+ mn edges mn edges by Theorem


55

3.1. By definitions definitions of the first, second and the third third leap Zagreb indices, indices, we get the followin followingg by applying Propositions 3. 3 .2, 3.3 and Observation 3. 3 .4.

  

LM 1 (T +11(G)) =

d2 (v/T +11(G))2

v∈V (T ( T +11(G))

d2 (v/T +11 (G))2 +

=

( G) v∈V (G

=

v∈V (G ( G)

LM 2 (T +11 (G)) =

− 1)2 + d1 (v/G) v/G)2 − 2(n 2(n − 1)d 1)d1 (v/G)] v/G)]

− 1)(n 1)(n2 − n − 4m) + M 1 (G).

     

d2 (u/T +11(G)) d2 (v/T +11 (G))

uv∈ uv∈E (T +11(G))





uv∈ uv∈E (G)

+

uv∈ uv∈E (L(G))

+

uv∈ / E (L(G))

+


+


= m(n

   





d2 (u/T +11(G)) d2 (v/T +11(G))

   

   

d2 (u/T +11(G)) d2 (v/T +11(G))

d2 (u/T +11(G)) d2 (v/T +11(G)) d2 (u/T +11(G)) d2 (v/T +11(G))

− 1)2 − (n − 1)M 1)M 1 (G) + M 2 (G).

   

d1 (v/T +11(G)) d2 (v/T +11(G))

v∈V (T ( T +11(G))

=



d2 (u/T +11(G)) d2 (v/T +11(G))

=

LM 3 (T +11 (G)) =

d2 (e/T +11(G))2

e∈E (G)

[(n [(n

= (n







d1 (v/T +11 (G)) d2 (v/T +11 (G))

( G) v∈V (G

 



d1 (e/T +11(G)) d2 (e/T +11(G))

+

e∈E (G)

= m[(n [(n

− 1)(n 1)(n + 2) − 2m] − M 1 (G).



Theorem 3.12 Let G be (n, m) graph. Then (1) LM (1) LM 1 (T −11 (G)) = M 1 (G); (2) LM (2) LM 2 (T −11 (G)) = M 2 (G); (3) LM (3) LM 3 (T −11 (G)) = 2m 2m(n + m

− 1) − M 1(G).

m (m−3) Proof Obviously, the graph T −11 (G) has n has n + m vertices and n(n2−1) + m( + mn edges, mn edges, 2 refer Theorem 3. 3 .1. Similarly, by definitions of the first, second and the third leap Zagreb indices

56


along with Propositions Propositions 3.2, 3.3 and Observation 3. 3 .4 we know the following.

 

LM 1(T −11 (G)) =

d2 (v/T −11 (G))2

11

v∈V ( V (T

−

(G))



d2 (v/T −11 (G))2 +

=

v∈V ( V (G)

e∈E (G)

= M 1(G).

LM 2 (T −11 (G)) =

         11

uv∈ uv∈E (T

−

d2 (u/T −11(G)) d2 (v/T −11 (G))

(G))

d2 (u/T −11(G))

=

uv∈ / E (G)

+







d2 (u/T −11(G)) d2 (v/T −11 (G))

 

 

d2 (u/T −11(G)) d2 (v/T −11 (G))

uv∈ / E (L(G))

+



d2 (v/T −11 (G))

uv∈ uv∈E (L(G))

+

d2 (e/T −11 (G))2

[d2 (u/T −11(G))][d ))][d2 (v/T −11 (G))]


+

[d2 (u/T −11(G))][d ))][d2 (v/T −11 (G))]


= M 2 (G).

LM 3 (T −11 (G)) =

   

v∈V ( V (T

11

−

=

d1 (v/T −11 (G))

(G))



d2 (v/T −11 (G))



d1 (v/T −11 (G)) d2 (v/T −11 (G))

v∈V ( V (G)

 



d1 (e/T −11(G)) d2 (e/T −11 (G))

+

e∈E (G)

= 2m(n + m

− 1) − M 1(G).



Theorem 3.13 Let G be (n, m) graph. Then (1) LM (1) LM 1 (T 0+1 (G)) = n( n (n

− 1)2 + m(m − 1)(m 1)(m + 3) − 2(m 2(m − 1)M 1)M 1 (G) + EM 1 (G); (2) LM (2) LM 2 (T 0+1 (G)) = [ (m−2 1) − n(n − 1)]M 1)]M 1 (G) − (m − 1)EM 1)EM 1 (G) + EM 2 (G) + m(m − 1)[n 1)[n(n − 1) − (m − 1)] + 2mn 2 mn((n − 1); (3) LM (3) LM 3 (T 0+1 (G)) = (m (m + n − 1)M 1)M 1(G) − EM 1 (G) + m[n(n + m) − 2(m 2(m − 1)]. 1)]. 2

Proof Notice that the graph T 0+1 (G) has n has n + m vertices and m and m((n 1) + M 12(G) edges by Theorem 3. 3.1. By definitions definitions of the first, second and the third leap Zagreb indices indices we get the

−


57

following by applying Propositions 3 .2, 3.3 and Observation 3. 3 .4.

   −

LM 1 (T 0+1(G)) =

d2 (v/T 0+1(G))2

v∈V (T (T 0+1(G))

v∈V (G (G)

=

(n

= n(n

2

− 1)

1)2 +

[(m [(m

e∈E (G)

+ m(m

          −  −  −

− 1)2 + d1(e/G) e/G)2 − 2(m 2(m − 1)d 1)d1 (e/G)] e/G)]

− 1)(m 1)(m + 3) − 2(m 2(m − 1)M 1)M 1 (G) + EM 1 (G).

d2 (u/T 0+1(G))



uv∈ uv∈E (T 0+1(G))

d2 (v/T 0+1(G))



d2 (u/T 0+1(G)) d2 (v/T 0+1(G))

=

  

uv∈ uv∈E (L(G))

+ +

(m

1)2

uv∈ uv∈E (L(G))

+

− (m − 1)(d 1)(d1 (u/G u/G)) + d1 (v/G)) v/G)) + d1 (u/G u/G)) · d1 (v/G) v/G)

[(n [(n

1)(m 1)(m

− 1 − d1(v/G))] v/G))]

[(n [(n

1)(m 1)(m

− 1 − d1(v/G))] v/G))]


+

 

d2 (u/T 0+1(G)) d2 (v/T 0+1(G))


=



d2 (u/T 0+1(G)) d2 (v/T 0+1(G))



=

d2 (e/T 0+1(G))2

e∈E (G)



v∈V (G (G)

LM 2 (T 0+1 (G)) =



d2 (v/T 0+1 (G))2 +

=

− 1)2 − n(n − 1)]M [ 1)]M 1 (G) − (m − 1)EM 1)EM 1 (G) + EM 2 (G) 2 +m(m − 1)[n 1)[n(n − 1) − (m − 1)] + 2mn 2mn((n − 1). 1). (m

LM 3 (T 0+1(G)) =

     −

d1 (v/T 0+1 (G)) d2 (v/T 0+1 (G))

v∈V ( V (T 0+1 (G))

=



d1 (v/T 0+1(G)) d2 (v/T 0+1(G))

v∈V ( V (G)

  



d1 (e/T 0+1(G)) d2 (e/T 0+1(G))

+

e∈E (G)

=



[m(n

1)] +

v∈V ( V (G)

= (m + n



[(n [(n + d1 (e/G))( e/G))(m m

e∈E (G)

− 1 − d1(e/G))] e/G))]

− 1)M 1)M 1 (G) − EM 1 (G) + m[n(n + m) − 2(m 2(m − 1)]. 1)].

Theorem 3.14 Let G be (n, m) graph. Then (1) LM (1) LM 1 (T 1+1 (G)) = m( m (m 1+1

(2) LM (2) LM 2 (T

(G)) =

− 1)(m 1)(m + 3) − 2(m 2(m − 1)M 1)M 1 (G) + EM 1 (G); − (m − 1)EM 1)EM 1 (G) + EM 2 (G) − m(m − 1)2 ;

(m−1)2 M 1 (G) 2



58


(3) LM (3) LM 3 (T 1+1 (G)) = (m (m

− n − 1)M 1)M 1(G) − EM 1 (G) + m[n(m + 1) − 2(m 2(m − 1)]. 1)].

Proof Clearly, the graph T graph T 1+1 (G) has n has n + m vertices and (n ( n 1)( n2 + m) + M 12(G) edges by Theorem 3. 3.1. By definitions of the first, second and the third leap Zagreb indices we therefore get the following by Propositions 3. 3 .2, 3.3 and Observation 3. 3 .4.

−

  

LM 1(T 1+1(G)) =

d2 (v/T 1+1(G))2

( T 1+1(G)) v∈V (T

d2 (v/T 1+1(G))2 +

=

v∈V (G ( G)

=

e∈E (G)

LM 2 (T 1+1 (G)) =

            −

− 1)2 + d1(e/G) e/G)2 − 2(m 2(m − 1)d 1)d1 (e/G) e/G)



− 1)(m 1)(m + 3) − 2(m 2(m − 1)M 1)M 1(G) + EM 1(G).

d2 (u/T 1+1(G))

uv∈ uv∈E (T 1+1(G))

 

uv∈ uv∈E (G)

+

d2 (u/T 1+1(G))

uv∈ / E (G)

+

d2 (v/T 1+1(G))



+

d2 (v/T 1+1(G))



  

  

d2 (u/T 1+1(G)) d2 (v/T 1+1(G))


+



d2 (u/T 1+1(G)) d2 (v/T 1+1 (G))

uv∈ uv∈E (L(G))

d2 (u/T 1+1(G)) d2 (v/T 1+1(G))


(m

1)2

uv∈ uv∈E (L(G))

=



d2 (u/T 1+1(G)) d2 (v/T 1+1(G))

=

=

d2 (e/T 1+1(G))2

e∈E (G)

(m

= m(m



(m

− (m − 1)(d 1)(d1 (u/G u/G)) + d1 (v/G)) v/G)) + d1 (u/G u/G)) · d1 (v/G) v/G)

− 1)2 M 1(G) − (m − 1)EM 1)EM 1 (G) + EM 2 (G) − m(m − 1)2 . 2

LM 3 (T 1+1(G)) =

   

d1 (v/T 1+1 (G)) d2 (v/T 1+1 (G))

( T 1+1(G)) v∈V (T

d1 (v/T 1+1 (G))

=

v∈V (G ( G)





d2 (v/T 1+1(G))

 



d1 (e/T 1+1 (G)) d2 (e/T 1+1(G))

+

e∈E (G)

= (m



− n − 1)M 1)M 1 (G) − EM 1 (G) + m[n(m + 1) − 2(m 2(m − 1)]. 1)].

Theorem 3.15 Let G be (n, m) graph. Then (1) LM (1) LM 1 (T ++1 (G)) = (n (n

− 1)[n 1)[n(n − 1) − 4m] + m( m(m − 1)(m 1)(m + 3)




59

− (2m (2m − 3)M 3)M 1 (G) + EM 1 (G); ++1

(2) LM (2) LM 2 (T

(G)) =



(m−1)2 2



− (n − 1)(n 1)(n + 1) M 1 (G) + M 2 (G) − (m − 1)EM 1)EM 1 (G) +EM 2(G) + m[n(2n (2n − 3) − m(3m (3m − 4) + mn( mn(n − 1)]

 

+

d2 (u/G u/G))d2 (v/G) v/G)

u∈V ( V (G),v∈ ,v ∈E (G),u∼ ,u∼v

+

d2 (u/G u/G))d2 (v/G); v/G);

u∈V ( V (G),v∈ ,v ∈E (G),u≁v

(3) LM (3) LM 3 (T ++1 (G)) = mn( mn(m + n

− 2) + (m (m − n − 2)M 2)M 1 (G) − EM 1 (G).

Proof Clearly, the graph T ++1(G) has n has n+ +m vertices and mn+ mn+ M 12(G) edges by Theorem 3.1. Now by definitions of the first, second and the third leap Zagreb indices, applying Propositions 3.2, 3.3 and Observation 3. 3 .4 we have the following.

   −

LM 1 (T ++1 (G)) =

d2 (v/T ++1(G))2

(T ++1(G)) v∈V (T

d2 (v/T ++1(G))2 +

=

v∈V (G (G)

=

[n

d2 (e/T ++1(G))2

e∈E (G)

1

(G) v∈V (G

= (n

 

− d1(v/G)] v/G)]2 +

[m

e∈E (G)

− 1 − d1(e/G)] e/G)]2

− 1)[n 1)[n(n − 1) − 4m] + m(m − 1)(m 1)(m + 3) − (2m (2m − 3)M 3)M 1 (G)

+EM 1 (G).

LM 2 (T ++1 (G)) =

    



uv∈ uv∈E (T ++1 (G))

d2 (u/T ++1(G)) d2 (v/T ++1 (G))



d2 (u/T ++1(G)) d2 (v/T ++1 (G))

=



uv∈ uv∈E (G)

+

uv∈ uv∈E (L(G))

+ +

(m

d2 (u/T ++1(G)) d2 (v/T ++1 (G))

− 1)2 − (n − 1)(n 1)(n + 1)

2 +EM 2 (G) + m[n(2n (2n +

   

 

− (m − 1)EM 1)EM 1 (G) − 3) − m(3m (3m − 4) + mn( mn(n − 1)] M 1 (G) + M 2 (G)

d2 (u/G u/G))d2 (v/G) v/G)

u∈V ( V (G),v∈ ,v∈E (G),u∼ ,u∼v

+

  

d2 (u/T ++1(G)) d2 (v/T ++1(G))




  



d2 (u/T ++1(G)) d2 (v/T ++1(G))


=



u∈V ( V (G),v∈ ,v∈E (G),u≁v

d2 (u/G u/G))d2 (v/G) v/G).

60


   

LM 3 (T ++1(G)) =

d1 (v/T ++1(G)) d2 (v/T ++1(G))

v∈V ( V (T ++1(G))





d1 (v/T ++1(G)) d2 (v/T ++1(G))

=

 

v∈V ( V (G)



d1 (e/T ++1(G)) d2 (e/T ++1(G))

+

e∈E (G)



− 2) + (m (m − n − 2)M 2)M 1 (G) − EM 1 (G).

= mn( mn(m + n

Theorem 3.16 Let G be (n, m) graph. Then (1) LM (1) LM 1 (T −+1 (G)) = m( m(m (2) LM 2 (T −+1(G)) = −



(m−1)2 2

− 1)(m 1)(m + 3) − (2m (2m − 3)M 3)M 1 (G) + EM 1 (G); M 1 (G) + M 2 (G) − (m − 1)EM 1)EM 1 (G) + EM 2 (G) + m(m − 1)(m 1)(m + 1)



d2 (u/G) u/G)d2 (v/G) v/G) +

u∈V ( V (G),v∈E (G),u∼v

(3) LM (3) LM 3 (T −+1 (G)) = (m (m





d2 (u/G) u/G)d2 (v/G) v/G) ;

u∈V ( V (G),v∈E (G),u≁ v

− n − 2)M 2)M 1 (G) − EM 1 (G) + mn( mn(m + 3). 3) .

Proof Notice that the graph T −+1 (G) has n has n + m vertices and n(n2−1) + m(n 2) + M 12(G) edges, refer Theorem 3. 3 .1. We are easily get the following by definitions of the first, second and the third leap Zagreb indices along with Propositions 3 .2, 3.3 and Observation 3. 3 .4.

−

 

LM 1(T −+1 (G)) =

v∈V ( V (T

d2 (v/T −+1 (G))2

+1

−

(G))

v∈V ( V (G)

= m(m LM 2 (T −+1(G))

=

    

uv∈E (T −+1 (G))

=



uv∈E (L(G))

+

uv∈E (S (G))

+

uv∈E (S(G))

=

e∈E (G)

d2 (u/T −+1 (G))

d2 (u/T −+1(G))

  





d2 (v/T −+1(G))

−



d2 (v/T −+1 (G))



d2 (u/T −+1 (G)) d2 (v/T −+1(G))

  

  

d2 (u/T −+1 (G))

d2 (v/T −+1 (G))

d2 (u/T −+1 (G))

d2 (v/T −+1 (G))

(m − 1)2 M 1 (G) + M 2 (G) − (m − 1)EM 1)EM 1 (G) 2 +EM 2 (G) + m(m − 1)(m 1)(m + 1)

 

d2 (e/T −+1 (G))2

− 1)(m 1)(m + 3) − (2m (2m − 3)M 3)M 1 (G) + EM 1 (G).

uv∈ / E (G)

+



d2 (v/T −+1 (G))2 +

=



u∈V ( V (G),v∈E (G),u∼v

d2 (u/G) u/G)d2 (v/G) v/G) +



u∈V ( V (G),v∈E (G),u≁ v

 

d2 (u/G) u/G)d2 (v/G) v/G) .


   

LM 3 (T −+1 (G)) =

61

d1 (v/T −+1 (G)) d2 (v/T −+1 (G))

v∈V ( V (T

+1

−



(G))



d1 (v/T −+1 (G)) d2 (v/T −+1 (G))

=

 

v∈V ( V (G)

d1 (e/T −+1(G))

+

e∈E (G)

= (m



d2 (e/T −+1 (G))



− n − 2)M 2)M 1 (G) − EM 1 (G) + mn( mn(m + 3). 3).

Theorem 3.17 Let G be (n, m) graph. Then (1) LM (1) LM 1 (T 0−1 (G)) = n( n(n

− 1)2 + EM 1(G);

(2) LM (2) LM 2 (T 0−1 (G)) = EM 2 (G) + n(n (3) LM (3) LM 3 (T 0−1 (G)) = (n (n + m

− 1)M 1)M 1 (G) − 2mn( mn(n − 1);

− 1)M 1)M 1 (G) − EM 1 (G) + m + m((n2 − 3n − 2m + 2). 2).

Proof Notice that the graph T 0−1 (G) has n has n + m vertices and m and m(( m2+1 + n) M 12(G ) edges, refer Theorem 3. 3 .1. By definitions definitions of the first, second and the third leap Zagreb indic indices es along with Propositions Propositions 3.2, 3.3 and Observation 3. 3 .4 we get the following.

−

LM 1(T 0−1 (G)) =

 

v∈V ( V (T 0

1

−

=

d2 (v/T 0−1 (G))2 (G))

d2 (v/T 0−1 (G))2 +

v∈V ( V (G)

LM 2 (T 0−1 (G)) =

d2 (e/T 0−1(G))2

e∈E (G)

− 1)2 + EM 1(G).

= n(n



          −

d2 (u/T 0−1 (G)) d2 (v/T 0−1 (G))

uv∈ uv∈E (T 0

1

−



(G))



d2 (u/T 0−1 (G)) d2 (v/T 0−1 (G))

=

  

uv∈ / E (L(G))

+

d2 (u/T 0−1(G)) d2 (v/T 0−1 (G))


+

[d1 (u/G u/G)) d1 (v/G)] v/G)] +

uv∈ / E (L(G))

+

 

d2 (u/T 0−1(G)) d2 (v/T 0−1 (G))


=



(n

·




1)d 1)d1 (v/G) v/G)


= EM 2 (G) + n(n

(n

− 1)M 1)M 1 (G) − 2mn( mn(n − 1). 1).

− 1)d 1)d1 (v/G) v/G)

62


LM 3 (T 0−1 (G)) =

   

v∈V ( V (T 0

1

−

=

d1 (v/T 0−1 (G)) d2 (v/T 0−1 (G))

(G))





d1 (v/T 0−1(G)) d2 (v/T 0−1 (G))

 

v∈V ( V (G)



d1 (e/T 0−1(G)) d2 (e/T 0−1 (G))

+

e∈E (G)

= (n + m



− 1)M 1)M 1 (G) − EM 1 (G) + m(n2 − 3n − 2m + 2). 2).

Theorem 3.18 Let G be (n, m) graph. Then

(1) LM (1) LM 1 (T 1−1 (G)) = EM 1 (G); (2) LM (2) LM 2 (T 1−1 (G)) = EM 2 (G); (3) LM (3) LM 3 (T 1−1 (G)) = (n (n + m

− 1)M 1)M 1 (G) − EM 1 (G) − 2m(n + m − 1). 1).

( 1) ( ) Proof Clearly, the graph T 1−1 (G) has n has n + m vertices m vertices and n n2− + m( m( m2+1 + n) M 12 G edges by Theorem 3. 3 .1. Whence, Whence, by definitions definitions of the first, second and the third leap Zagreb indices indices along with Propositions Propositions 3 .2, 3.3 and Observation 3. 3 .4 we get the following.

−

LM 1(T 1−1 (G)) =

 

v∈V ( V (T 1

1

−

=

d2 (v/T 1−1 (G))2 (G))

d2 (v/T 1−1 (G))2 +

v∈V ( V (G)

         

uv∈ uv∈E (T 1

1

−

d2 (u/T 1−1(G)) d2 (v/T 1−1 (G))

(G))

d2 (u/T 1−1(G))

=

uv∈ uv∈E (G)

+

d2 (v/T 1−1 (G))

 





  

  

d2 (u/T 1−1 (G)) d2 (v/T 1−1 (G))


+



d2 (u/T 1−1(G)) d2 (v/T 1−1(G))

uv∈ / E (L(G))

+



d2 (u/T 1−1 (G)) d2 (v/T 1−1 (G))

uv∈ / E (G)

+

d2 (e/T 1−1(G))2

e∈E (G)

= EM 1 (G).

LM 2 (T 1−1(G)) =



d2 (u/T 1−1 (G)) d2 (v/T 1−1 (G))


= EM 2 (G).


   

LM 3(T 1−1 (G)) =

v∈V ( V (T 1

1

−

d1 (v/T 1−1 (G)) d2 (v/T 1−1 (G))



(G))



d1 (v/T 1−1 (G)) d2 (v/T 1−1 (G))

=

 

v∈V ( V (G)

d1 (e/T 1−1(G))

+

e∈E (G)

= (n + m

63



d2 (e/T 1−1(G))



− 1)M 1)M 1 (G) − EM 1 (G) − 2m(n + m − 1). 1).

Theorem 3.19 Let G be (n, m) graph. Then (1) LM (1) LM 1 (T +−1 (G)) = M 1 (G) + EM 1 (G) + (n (n

− 1)[n 1)[n(n − 1) − 4m];

(2) LM 2 (T +−1(G)) = (n (n − 1)2 M 1 (G) + M 2 (G) + EM 2 (G) − m(n − 1)(n 1)(n + 1) −





d2 (u/G) u/G)d2 (v/G) v/G) +

u∈V ( V (G),v∈E (G),u∼v

(3) LM (3) LM 3(T +−1 (G)) = (n (n + m





d2 (u/G) u/G)d2 (v/G) v/G) ;

u∈V ( V (G),v∈E (G),u≁ v

− 2)M 2)M 1 (G) − EM 1 (G) + m[(n [(n − 1)(n 1)(n + 2) − 2(2m 2(2m + n − 1)]. 1)].

Proof Clearly, the graph T +−1 (G) has n has n + + m m vertices vertices and m( m ( m2+3 + n) n) M 12(G) edges by Theorem 3. 3.1. By definitions definitions of the first, second second and the third leap Zagreb Zagreb indices indices along with Propositions 3. 3.2, 3.3 and Observation 3. 3 .4 we therefore get the following.

−

 

LM 1(T +−1 (G)) =

v∈V ( V (T +

d2 (v/T +−1 (G))2 1

−

(G))

v∈V ( V (G)

=

    

uv∈E (T +−1 (G))

=



d2 (u/T +−1 (G))

d2 (u/T +−1 (G))

uv∈E (G)

+

uv∈ / E (L(G))

+

uv∈E (S (G))

+

uv∈E (S(G))

=

  

d2 (e/T +−1(G))2

e∈E (G)

= M 1 (G) + EM 1 (G) + (n (n LM 2 (T +−1(G))



d2 (v/T +−1 (G))2 +

=





− 1)[n 1)[n(n − 1) − 4m].

d2 (v/T +−1 (G))



d2 (v/T +−1 (G))



d2 (u/T +−1 (G)) d2 (v/T +−1(G))

  

  

d2 (u/T +−1 (G))

d2 (v/T +−1 (G))

d2 (u/T +−1 (G))

d2 (v/T +−1 (G))

(n − 1)2 M 1 (G) + M 2 (G) + EM 2 (G) − m(n − 1)(n 1)(n + 1) −

 



u∈V ( V (G),v∈E (G),u∼v

d2 (u/G) u/G)d2 (v/G) v/G) +



u∈V ( V (G),v∈E (G),u≁ v

 

d2 (u/G) u/G)d2 (v/G) v/G) .

64


LM 3 (T +−1(G)) =

   

v∈V (T (T +

1

−

=

d1 (v/T +−1(G))



(G))

d2 (v/T +−1 (G))



d1 (v/T +−1 (G)) d2 (v/T +−1(G))

 

v∈V (G (G)



d1 (e/T +−1 (G)) d2 (e/T +−1 (G))

+

e∈E (G)

= (n + m



− 2)M 2)M 1 (G) − EM 1 (G) + m[(n [(n − 1)(n 1)(n + 2) − 2(2m 2(2m + n − 1)]. 1)].

Theorem 3.20 Let G be (n, m) graph. Then −−1 (1) LM (1) LM 1 (T −−1 (G)) = M 1 (G) + EM 1 (G); −−1 (2) LM (2) LM 2 (T −−1 (G)) = M 1 (G) + EM 2 (G) +



d2 (u/G u/G))d2 (v/G) v/G)

u∈V ( V (G),v∈ ,v ∈E (G),u∼ ,u∼v



+

d2 (u/G u/G))d2 (v/G); v/G);

u∈V ( V (G),v∈ ,v ∈E (G),u≁v −−1 (3) LM (3) LM 3 (T −−1 (G)) = (n (n + m

− 2)M 2)M 1 (G) − EM 1 (G).

−−1 Proof Notice that the graph T −−1 (G) has n has n + m vertices and n(n2−1) + m( m2−1 + n) M 12(G) edges by Theorem 3. 3 .1. By definit definition ionss of the first, first, second second and the third third leap leap Zag Zagreb reb indices indices,, Propositions 3. 3.2, 3.3 and Observation 3. 3 .4, we are easily get the following.

−

−−1 LM 1 (T −−1 (G)) =

 

v∈V (T ( T

−−1 d2 (v/T −−1 (G))2 1

−−

=

(G))

−−1 d2 (v/T −−1 (G))2 +

v∈V (G ( G)

    

uv∈ uv ∈E (T

1

−−

(G))



−−1 −−1 d2 (u/T −−1 (G)) d2 (v/T −−1 (G))

−−1 d2 (u/T −−1 (G))

=



uv ∈ / E (G)

+

uv∈ / E (L(G))

+

  



−−1 d2 (v/T −−1 (G))



−−1 −−1 d2 (u/T −−1 (G)) d2 (v/T −−1 (G))

   

  

−−1 −−1 d2 (u/T −−1 (G)) d2 (v/T −−1 (G))


= M 1 (G) + EM 2 (G) +

u∈V ( V (G),v∈ ,v ∈E (G),u∼ ,u∼v

+



−−1 −−1 d2 (u/T −−1 (G)) d2 (v/T −−1 (G))


+

−−1 d2 (e/T −−1 (G))2

e∈E (G)

= M 1 (G) + EM 1 (G).

−−1 LM 2 (T −−1 (G)) =





u∈V ( V (G),v∈ ,v ∈E (G),u≁v

d2 (u/G u/G))d2 (v/G) v/G).

d2 (u/G u/G))d2 (v/G) v/G)


−−1 LM 3 (T −−1 (G)) =

   

v∈V ( V (T

1

−−

=

−−1 −−1 d1 (v/T −−1 (G)) d2 (v/T −−1 (G))

(G))





−−1 −−1 d1 (v/T −−1 (G)) d2 (v/T −−1 (G))

v∈V ( V (G)

65

 



−−1 −−1 d1 (e/T −−1 (G)) d2 (e/T −−1 (G))

+

e∈E (G)

= (n + m

− 2)M 2)M 1 (G) − EM 1 (G).



References [1] A. R. Ashrafi, T. Doˇslić, A. Hamzeh, The Zagreb coindices of graph operations, Discrete Appl. Math., 158(15), 158(15), 1571–1578, 2010. [2] W. Baoy Baoyindur indureng, eng, M. Jixiang, Jixiang, Basic properties properties of total transformat transformation ion graphs, J. Math. Study , 34(2), 34(2), 109–116 109–116,, 2001. [3] B. Basavanagoud, Basavanagoud, Basic properties of generalized xyz-Point-Line transformation graphs , J. Inf. Optim. Sci., 39(2), 561–580, 2018, DOI: 10.1080/02522667.2017.1395147. [4] B. Basavanagoud, C. S. Gali, Computing first and second Zagreb indices of generalized transformation graphs graphs , J. Globa xyz -Point-Line transformation Globall Resear Research ch Math. Math. Arch., Arch., 5(4), 5(4), 100– 100–122 122,, 2018. [5] C. M. Da fonseca, fonseca, D. Stevanov Stevanovi´ ić, Further urther properties properties of the second Zagreb index, MATCH Commun. Math. Comput. Chem. , 72, 655–668, 2014. [6] A. Deng, A. Kelmans, J. Meng, Laplacian Laplacian Spectra of regular regular graph transforma transformation tions, s, Discrete Appl. Math., 161, 118–133, 2013. [7] B. Furtula, urtula, I. Gutman, Gutman, M. Dehmer, Dehmer, On structurestructure-sensi sensitivi tivity ty of degree-base degree-based d topological topological Appl. Math. Math. Comput. Comput., 219, 8973–8978, 2013. indices, Appl. [8] M. Goubko, T. Réti, Note on minimizing degree-based topological indices of trees with MATCH Commun. Commun. Math. Math. Comput. Comput. Chem. Chem. , 72, 633– given number of pendent vertices, MATCH 639, 2014. [9] I. Gutman, Gutman, N. Trinajsti´ Trinajstić, Graph theory and molecular orbitals, Total π-electron π -electron energy of alternate hydrocarbons, Chem. Phys. Lett. , 17, 535–538, 1972.

Commun. Math. [10] I. Gutman, K. C. Das, The first Zagreb index 30 years after, MATCH Commun. Comput. Chem., 50, 83–92, 2004. [11] F. Harary, Harary, Graph Theory , Addison-Wesely, Reading Mass, 1969. [12] S. M. Hosamani, Hosamani, N. Trinajstić, On reformulated Zagreb coindices, Research Gate , 2015-0 2015-05508 T 09:07:00 UTC. [13] V. R. Kulli, College Graph Theory , Vishwa International Publications, Gulbarga, India, 2012. [14] A. Mili´ Mili´ cevi´ c, S. Nikolić, N. Trinajstić, On reformulated Zagreb indices, Mol. Divers., 8(4), 393–399, 2004. [15] S. Nikolić, G. Kovaˇcevi´ c, A. Milićevi´ c, N. Trinajstić, The Zagreb indices 30 years after, Croat. Chem. Acta. , 76(2), 76(2), 113–12 113–124, 4, 2003.

66


[16]] A. M. Naji, N. D. Soner, Ivan [16 Ivan Gutman Gutman,, On Leap Zag Zagreb reb indices indices of graphs, graphs, Commun. Comb. Optim., 2(2), 99–117, 2017. Int.. J. [17] N. D. Soner, A. M. Naji, The k-distance neighbourhood polynomial of a graph, Int Math. Math. Comput. Comput. Sci. WASET Confere Conferenc ncee Proc Procee eeding dings, s, San Franci Francico co,, USA, Sep 26-27, 3(9) Part XV, 2359–2364, 2016. [18] G. Su, L. Xiong, L. Xu, The Nordhaus-Gaddum-type inequalities inequalities for the Zagreb index and Appl. Math. Math. Lett. Lett., 25(11), coindex of graphs, Appl. 25(11), 1701–17 1701–1707, 07, 2012.


A Generalization on Product Degree Distance of Strong Product of Graphs K.Pattabiraman (Department of Mathematics, Annamalai University, Annamalainagar 608 002, India) E-mail: [email protected]

Abstract: Abstract: In this this paper, paper, the exact exact formu formulae lae for the gener generali alize zed d product product degree degree disdistance tance,, recipr reciproca ocall product product degree degree distan distance ce and product product degre degreee distan distance ce of strong strong prodproduct of a connected graph and the complete multipartite graph with partite sets of sizes m0 , m1 , · · · , mr−1 are obtained.

Key Words: Words: Reciprocal product degree distance, product degree distance, strong product. AMS(2010): AMS(2010): 05C12, 05C76

§1. Introduction

All the graphs considered in this paper are simple and connected. For vertices u, v V ( V (G), the distance between u and v in G, denoted G, denoted by dG (u, v ), is the length of a shortest (u, ( u, v)-path in G and let dG (v) be the degree of a vertex v V ( V (G). The strong product of graphs G and H, denoted by G by G ⊠ H, is H, is the graph with vertex set V ( V (G) V ( V (H ) = (u, v ) : u V ( V (G), v V ( V (H ) and (u, (u, x)(v, )(v, y ) is an edge whenever (i ( i) u = v = v and xy E (H ), or (ii) ii) uv E (G) and x and x = = y y,, or (iii) iii) uv E (G) and xy and xy E (H ).

∈

∈

∈

× ∈

∈

{

∈

∈

∈

}

A topological index of a graph is a real number related to the graph; it does not depend on labeling or pictorial representation of a graph. In theoretical chemistry, molecular structure descriptors (also called topological indices) are used for modeling physicochemical, pharmacologic, cologic, toxicolog toxicologic, ic, biological biological and other properties properties of chemica chemicall compounds [12]. There exist several several types of such indices, especially especially those based on vertex vertex and edge distances. distances. One of the most intensively studied topological indices is the Wiener index. Let G Let G be a connected graph. Then Wiener index of index of G G is defined as W ( W (G) =

1 2



dG (u, v )

u, v ∈ V ( V (G)

with the summation going over all pairs of distinct vertices of G. of G. This This definition can be further

1 Received


68

K.Pattabiraman

generalized in the following way: W λ (G) =

1 2



dλG (u, v ),

u, v ∈ V (G (G)

where d where d λG (u, v ) = (dG (u, v ))λ and λ and λ is is a real number [13, 14]. If λ If λ = = 1, then W then W −1 (G) = H ( H (G), where H where H ((G) is Harary index of G. In G. In the chemical literature also W 12 [29] as well as the general case W case W λ were examined [10, 15].

−

Dobrynin and Kochetova [6] and Gutman [11] independently proposed a vertex-degreeweighted version of Wiener index called degree distance , which is defined for a connected graph G as 1 DD((G) = DD (dG (u) + dG (v))d ))dG (u, v ), 2



u,v∈ u,v∈V (G (G)

where d where dG (u) is the degree of the vertex u in u in G. G. Similarly, Similarly, the product degree distance or Gutman index of of a connected graph G is defined as DD∗ (G) =

1 2



dG (u)dG (v)dG (u, v ).

u,v∈ u,v∈V (G (G)

The additively weighted Harary index (H A ) or reciprocal degree distance (RDD) RDD ) is defined in [3] as 1 (dG (u) + dG (v)) H A (G) = RDD( RDD (G) = . 2 dG (u, v )



u,v∈ u,v∈V ( V (G)

Similarly Similarly,, Su et al. [28] introduce introduce the reciprocal product degree distance of graphs, which can be seen as a product-degr product-degree-w ee-weigh eightt version of Harary index RDD∗ (G) =

1 2



u,v∈ u,v∈V ( V (G)

dG (u)dG (v) . dG (u, v )

In [16], Hamzeh et al. recently recently introduced introduced generalized generalized degree distance distance of graphs. graphs. Hua and Zhang [18] have obtained lower and upper bounds for the reciprocal degree distance of graph in terms of other graph invariants. invariants. Pattabiraman et al. [22, 23] have have obtained the reciprocal degree distance of join, tensor product, strong product and wreath product of two connected graphs in terms of other graph invariants. invariants. The chemical applications applications and mathematical properties of the reciprocal degree distance are well studied in [3, 20, 27].

The generalized degree distance , denoted by H by H λ (G), is defined as H λ (G) =

1 2



(dG (u) + dG (v))d ))dλG (u, v ),

u,v∈ u,v∈V ( V (G)

where λ is a real real numb number er.. If λ = 1, then H λ (G) = DD DD((G) and if λ =

−1, then H λ(G) =

69

A Generalization on Product Degree Distance of Strong Product of Graphs

RDD( RDD (G). Similarly, generalized product degree distance , denoted by H by H λ∗ (G), is defined as H λ∗ (G) =

1 2



dG (u)dG (v)dλG (u, v ).

u,v∈ u,v ∈V (G ( G)

If λ = 1, then H λ∗ (G) = DD∗ (G) and if λ = 1, then H λ∗ (G) = RDD∗ (G). Therefore the study of the above topological indices are important and we try to obtain the results related to these indices. indices. The generalized generalized degree distance of unicyclic unicyclic and bicyclic bicyclic graphs are studied by Hamzeh Hamzeh et al. [16, 17]. Also they are given the generaliz generalized ed degree distance distance of Cartesian Cartesian product, product, join, symmetric symmetric difference, difference, composition composition and disjuncti disjunction on of two graphs. The generalized degree distance and generalized product degree distance of some classes of graphs are obtained obtained in [24, 25, 26]. In this paper, the exact formulae formulae for the generalize generalized d product degree distance, reciprocal product degree distance and product degree distance of strong product G ⊠ K m0 , m1 , ··· , mr 1 , where K m0 , m1 , ··· , mr 1 is the complete multipartite graph with partite sets of sizes m 0 , m1 , , mr −1 are obtained.

−

−

−

·· · · ·

The first Zagreb index is defined as



M 1 (G) =

dG (u)2

(G) u∈V (G

and the second Zagreb index is defined as



M 2 (G) =

dG (u)dG (v).

uv∈ uv∈E (G)

In fact, one can rewrite the first Zagreb index as M 1 (G) =



(dG (u) + d + dG (v)). )).

uv∈ uv∈E (G)

The Zagreb indices were found to be successful in chemical and physico-chemical applications, tions, especially especially in QSPR/QSAR QSPR/QSAR studies, studies, see [8, 9].

⊆ ⊆

 

⊂ ⊂

For S or S V ( V (G), S denotes the subgraph of G induced G induced by S. by S. For For two subsets S, subsets S, T V ( V (G), not necessarily disjoint, by d G (S, T ) T ), we mean the sum of the distances in G from each vertex of S to S to every vertex of T of T , that is, d is, d G (S, T ) T ) = dG (s, t).



s ∈ S, t ∈ T

§2. Generalized Generalized Product Product Degree Distance Distance of Strong Strong Product of Graphs

In this section, we obtain the Generalized product degree distance of G of G ⊠ K m0 , m1 , ··· , mr 1 . Let let K m0 , m1 , ··· , mr 1 , r 2, 2 , G be a simple connected graph with V ( V (G) = v0 , v1 , , vn−1 and let K be the complete multiparite graph with partite sets V 0 , V 1 , , V r−1 and let V i = m i , 0 i r 1. In the graph G ⊠ K m0 , m1 , ··· , mr 1 , let B let B ij = v i V j , vi V ( V (G) and 0 j r 1. −

{

≤ −

−

···

} ·· · · · × ∈

≥ | | ≤ ≤ ≤ − −

70

K.Pattabiraman

For our convenience, the vertex set of G of G ⊠ K m0, m1 , ··· , mr 1 is written as −

r −1 n−1

V ( V (G)

× V ( V (K m , m ,..., ,..., m 0

{ }ij ==00,,11,,······,, nr−−11. Let X Let X i =

Let B = Bij

1

)= 1

r−



Bij .

i=0 j = 0

r−1

n−1



Bij and Y and Y j =

j = 0



Bij ; we call X i and Y and Y j as layer as layer

i=0

{

···

}

and and column of G ⊠ K m0 , m1 , ...,m respectively y. If we denote V ( V (Bij ) = xi1 , xi2 , , ximj ...,m r 1 , respectivel and V and V ((Bkp ) = xk1 , xk2 , , xk mp , then xiℓ and x and x kℓ , 1 ℓ j, are called the corresponding vertices of of B Bij and B and Bkp . Further, if v if v i vk E (G), then the induced subgraph Bij Bkp of G G ⊠ K m0 , m1 , ··· , mr 1 is isomorphic to K |V j ||V or, m p independent edges joining the corresponding ||V p | or, m vertices of B of B ij and B and B kj according as j as j = p or p or j = p, respectively. The following remark is follows from the structure of the graph K m0 , m1 , ··· , mr 1 . −

{

···

}

≤ ≤

∈



−







−

Remark 2.1 Let n0 and q be the number of vertices and edges of K m0 , m1 , ··· , mr 1 . Then the sums −

r −1



mj m p

= 2q,

j, p = 0 j  = p r−1



mj2

= n20

j =0

r−1



mj2 m p = n = n 0 q

j, p = 0 j  = p

− 2q,

r−1

− 3t



=

mj m p2 ,

j, p = 0 j  = p

r−1



mj3

= n30

j =0

and

− 3n0q + + 3t 3t

r−1



mj4 = n 40

j=0

− 4n20q + + 2q 2 q 2 + 4n 4n0t − 4τ, ′

where t and τ τ are the number of triangles and K 4s in K K m0 , m1 , ··· , mr 1 . −

The proof of the following lemma follows easily from the properties and structure of G ⊠ K m0 , m1 , ··· , mr 1 . −

be a conne connecte cted d graph graph and let Bij , Bkp Lemma 2.2 Let G be K m0 , m1 , ··· , mr 1 , where r 2. 2 . Then −

≥

(i) If vi vk

∈ E (G) and xit ∈ Bij , xkℓ ∈ Bkj , then dG (xit , xkℓ ) = ′

 

1, if t = ℓ, = ℓ, 2, if t = ℓ,



∈

B of

the graph graph G′ = G

⊠

71


and if xit

∈ Bij , xkℓ ∈ Bkp, j = p, then dG (xit, xkℓ ) = 1. then for for any two vertic vertices es xit ∈ (ii) ii) If vi vk ∈ / E (G), then ′

Bij , xkℓ

dG (vi , vk ).

∈

Bkp , dG (xit , xkℓ ) = ′

(iii) iii) For any two distinct vertices in Bij , their distance is 2. The proof of the following lemma follows easily from Lemma 2 .2, which is used in the proof of the main theorems of this section.

be a conne connecte cted d graph graph and let Bij , Bkp Lemma 2.3 Let G be K m0 , m1 , ...,m r 2. 2 . ...,m r 1 , where r

≥

−

∈

the graph graph G′ = G

B of

⊠

(i) If vi vk

∈ E (G), then dH G (Bij , Bkp ) = ′

(ii) ii) If v i vk / E (G), then

∈

 

dH G (Bij , Bkp ) = ′

(iii) iii) dH G (Bij , Bip ) = ′

 

mj m p, if j = p, mj (mj +1) , 2

 



if j = p,

mj mp dG (vi ,vk ) , m2j dG (vi ,vk ) ,

if j = p,



if j = p.



mj m p , if j = p, mj (mj −1) , 2

if j = p. = p.

Lemma 2.4 Let G be a connected graph and let Bij in G′ = G degree of a vertex (vi , uj ) Bij in G G ′ is

⊠

K m0 , m1 , ··· , mr 1 . Then the −

∈

dG ((v ((vi , uj )) = dG (vi ) + (n (n0 ′

where n0 =

− mj ) + dG(vi )(n )(n0 − mj ),

r−1



j=0

mj .

Now we obtain obtain the generalized generalized product degree degree distance distance of G of G ⊠ K m0 , m1 , ··· , mr 1 . −

Theorem 2.5 Let G G be a connected graph with n vertices and m edges. Then H λ∗ (G ⊠ K m0 , m1 , ··· , mr 1 ) −

= (4q (4 q 2 + n20 + 4n 4 n0 q )H λ∗ (G) + 4q 4 q 2 W λ (G) + (4q (4 q 2 + 2n 2n0 q )H λ (G) + M 1 (G) + 4n20 q 2



2

3n20

n (4q (4q 2 2

− n0q − 3t)



2n30 + 8τ 8 τ

− 2q + 4n − 4n0 t + 9t 9t + 7n 7n0 q − n0 − +m 3n0 q + + 2n 2 n0 t − 2q 2 − 3t − 4q + + 4τ 4 τ +2λ M 1 (G)(2q )(2q 2 − 2n0 t − 6t − 2q − 4τ ) τ ) + m(2q (2q 2 − 2n0 t − n0 q − 3t − 4τ ) τ ) +(2λ − 1)M 1)M 2 (G) 2q 2 − 2n0 t − 3n30 + 10n 10 n0 q + n + n20 − 18 18tt − 6q − n0 − 4τ .

 









72

K.Pattabiraman

Proof Let G Let G ′ = G ⊠ K m0 , m1 , ...,m ...,m r 1 . Clearly, −

1 2

H λ∗ (G′ ) =

1 2

=

     

dG (Bij )dG (Bkp )dλG (Bij , Bkp ) ′

′

′

Bij , Bkp ∈ B n−1 r−1

dG (Bij )dG (Bip )dλG (Bij , Bip ) ′

′

′

i = 0 j, p = 0 = p j 

n−1

r−1

dG (Bij )dG (Bkj )dλG (Bij , Bkj )

+

′

′

′

i, k = 0 j = 0 =k i n−1

r −1

dG (Bij )dG (Bkp )dλG (Bij , Bkp )

+

′

′

′

i, k = 0 j, p = 0 = p = k j  i n−1 r−1



dG (Bij )dG (Bij )dλG (Bij , Bij ) .

+

′

′

′

i = 0 j = 0

(2. (2.1)

We shall obtain the sums of (2. (2 .1) are separately.

First we calculate A calculate A 1 =

n−1 r−1



i = 0 j, p = 0 j  = p

By Lemma 2. 2.4, we have T 1′

dG (Bij )dG (Bip )dλG (Bij , Bip ). For that first we find T 1′ . ′

′

′

= dG (Bij )dG (Bip ) ′

 

′





− mj + 1) + (n ( n0 − mj ) dG (vi )(n )(n0 − m p + 1) + (n ( n0 − m p ) 2 (n0 + 1) 2 − (n0 + 1)m 1) mj − (n0 + 1)m 1) m p + mj m p dG (vi ) + 2n0 (n0 + 1) − (2n (2n0 + 1)m 1) mj − (2n (2n0 + 1)m 1) m p + 2m 2 mj m p dG (vi ) + n20 − n0 m p − n0 mj + mj m p .

= =

dG (vi )(n )(n0

 







From Lemma 2. 2 .3, we have d have d λG (Bij , Bip ) = m j m p. Thus ′

T 1′ dλG (Bij , Bip ) = T 1′ mj m p ′

  

2

= (n0 + 1) mj m p

−

(n0 + 1)m 1) mj2 m p

−

(n0 + 1)m 1) mj m p2 +



2 dG (vi )

− (2n (2n0 + 1)m 1) mj2 m p − (2n (2n0 + 1)m 1) mj m p2 + 2m 2 mj2 m p2 n20 mj m p − n0 mj2 m p − n0 mj m p2 + mj2 m p2 .

+ 2n0 (n0 + 1)m 1) mj m p +

mj2 m p2





dG (vi )

73


By Remark 2. 2.1, we have r−1

T 1

   

=

T 1′ dλG (Bij , Bip ) ′

j, p = 0 j  = p



2 2q 2 + 2qn 2 qn0 + 2n 2 n0 t + 2q 2q + + 4τ 4 τ + + 6t 6 t dG (vi )

=

+ 2qn0 + 4n 4 n0 t



2

− 4q + 6t 6t + 8τ 8τ dG (vi )



2

+ 2n0 t + 2q 2 q + 4τ 4τ .

From the definition of the first Zagreb index, we have n−1

A1

=

     T 1

i=0

=

 

2q 2 + 2qn 2qn 0 + 2n 2 n0 t + 2q 2q + + 4τ 4 τ + + 6t 6 t M 1 (G)

+2m +2m 2qn0 + 4n 4 n0 t

− 4q 2 + 6t 6t + 8τ 8τ



2

+n 2n0 t + 2q 2q + 4τ 4τ .

n−1

Next we obtain A2 =

r−1

i, k = 0 j = 0 =k i

By Lemma 2. 2.4, we have T 2′

dG (Bij )dG (Bkj )dλG (Bij , Bkj ). For that first we find T 2′ . ′

′

′

= dG (Bij )dG (Bkj ) ′

=

′



dG (vi )(n )(n0



− mj + 1) + (n ( n 0 − mj )

dG (vk )(n )(n0

− mj + 1) + (n ( n0 − mj )



− mj + 1) 2dG(vi )dG(vk ) + (n (n0 − mj )(n )(n0 − mj + 1)(d 1)(dG (vi ) + dG (vk )) +(n +(n0 − mj )2 .

= (n0

Thus r−1

A2

=

n−1

  

T 2′ dλG (Bij , Bkj ) ′

j = 0 i, k = 0 i =k r−1

=

j = 0

n−1

i, k = 0 i =k vi vk ∈E (G)

r−1

T 2′ dλG

′

(Bij , Bkj ) +

n−1

 

j = 0

i, k = 0 i =k vi vk ∈ / E (G)

T 2′ dλG (Bij , Bkj ) ′

74

K.Pattabiraman

By Lemma 2. 2 .3, we have r−1

A2

n−1

  − T 2′

=

j = 0

i, k = 0 i =k vi vk ∈E (G)

r−1

n−1

1

r−1

=

j = 0

i, k = 0 =k i vi vk ∈E (G)

r−1

n−1

 

=

j = 0

1

T 2′ (2λ

i, k = 0 i =k vi vk ∈E (G)

n−1

  

λ

T 2′ mj2 dλG (vi , vk ),

2 + 2 mj mj +

   − T 2′

λ

j = 0

λ

r−1



λ

2 + 2 mj mj +



− 1)

i, k = 0 i =k vi vk ∈ / E (G)

mj2

−

mj2

n−1

  

T 2′ mj2 dλG (vi , vk )

+

j = 0

i, k = 0 =k i vi vk ∈ / E (G)

r−1 n−1

mj2

− mj

  +

T 2′ mj2 dλG (vi , vk )

j = 0 i, k = 0 i =k

= S 1 + S 2 ,

(2. (2.2)

where S where S 1 and S and S 2 are the sums of the terms of the above expression, in order. Now we calculate S 1 . For that first we find the following. (2λ

− 1)T 1)T 2′



mj2

− mj





= (2λ



mj4

− 1)

− (2n (2n0 + 3)m 3) mj3 + (n20 + 4n 4 n0 + 3)m 3) mj2

−(n0 + 1) 2mj dG(vi )dG(vk ) + mj4 − (2n (2n0 + 2)m 2) mj3 + (n ( n20 + 3n 3 n0 + 1)m 1) mj2 − (n20 + n0 )mj + mj4 − (2n (2n0 + 1)m 1) mj3 + (n ( n20 + 2n 2 n0 )mj2 − no2 mj .

 





(dG (vi ) + dG (vk ))


T 2′′



=

(2

λ

j = 0

= (2

λ

−



1)

− −  −

+ 2q 2

1)T 1)T 2′

+ 2q 2



mj2

− mj

2

 3n30 + 10n 10 n0q

2q

18 18tt + n20

− 2n0t − 4τ − − − 4τ − − 2n0t − 6t − 2q (dG(vi ) + dG(vk )) − 2n0t − n0q − 3t . 4τ −

− 6q − n0

 



dG (vi )dG (vk )

Hence n−1

S 1



=

T 2′′

i, k = 0 i =k vi vk ∈E (G) λ



2

 − − − − −  − − −

3n30 + 10n 10 n0 q

− 4τ − − − 18 18tt + 2q 2 4τ 2n0 t − 6t − 2q 2M 1 (G) +2m +2m 2q 2 4τ 2n0 t − n0 q − 3t .

= (2

1)

2q

2n0 t





+ n20

− 6q − n0



2M 2 (G)

75


Next we calculate S 2 . For that we need the following. T 2′ mj2



=

mj4

(2n (2n0 + 2)m 2) mj3 + (n ( n0 + 1) 2 mj2

 − −  − mj4

+

+ mj4

(2n (2n0 + 1)m 1) mj3 +

(n20 +



dG (vi )dG (vk )

n0 )mj2



2n0mj3 + n20 mj2 .



(dG (vi ) + dG (vk ))


T 2

=

  − − − −  − − − −  − − −  T 2′ mj2

j = 0

=

2q 2

4τ

2n0 t

6t + 2n 2n0 q

+ 2q 2

4τ

2n0 t

+ 2q 2

4τ

2n0 t .

− 2q + + n20





dG (vi )dG (vk )

3t + n0 q (dG (vi ) + dG (vk ))

From the definitions of H of H λ∗ ,H λ and W and W λ , we obtain n−1

S 2

  − − − −  − − − −  − − −    T 2 dλG (vi , vk )

=

i, k = 0 =k i

2

= 2 2q

4τ

2n0 t

6t + 2n 2n0 q

+2 2q 2

4τ

2n0 t

+2 2q 2

4τ

2n0 t W λ (G).

n−1

Now we calculate A calculate A 3 =

r−1



3t + n0 q H λ (G)



H λ∗(G)

dG (Bij )dG (Bkp )dλG (Bij , Bkp ). For that first we com′

i, k = 0 j, p = 0 = p i = k j 

− 2q + +

n20

′

′

pute T pute T 3′ . By Lemma 2. 2 .4, we have T 3′

= dG (Bij )dG (Bkp ) ′

= =



dG (vi )(n )(n0

′





− mj + 1) + (n ( n0 − mj ) dG (vk )(n )(n0 − m p + 1) + (n (n0 − m p ) dG (vi )dG (vk )(n )(n0 − mj + 1)(n 1)(n0 − m p + 1) + dG (vi )(n )(n0 − mj + 1)(n 1)( n0 − m p ) +dG (vk )(n )(n0 − m p + 1)(n 1)(n0 − mj ) + (n (n0 − mj )(n )(n0 − m p ).

76

K.Pattabiraman

Since the distance between B ij and B and B kp is m is m j m p dλG (vi , vk ). Thus T 3′ mj m p



= dG (vi )dG (vk )

 

+dG (vi )

(n20 + 2n 2 n0 + 1)m 1) mj m p

(n20 +

(n0 + 1)m 1) mj2 m p

−

(n0 + 1)m 1) mj m p2

n0 )mj m p

−

(n0 + 1)m 1) mj m p2 +

n0 mj2 m p +

mj2 m p2

mj2 m p2

 

− − +dG (vk ) (n20 + n0 )mj m p − n0 mj m p2 − (n0 + 1)m 1) mj2 m p + mj2 m p2 + n20 mj m p − n0 mj m p2 − n0 mj2 m p + mj2 m p2 .







By Remark 2. 2.1, we obtain r−1

T 3



=





T 3′ mj m p = dG (vi )dG (vk ) 2n0 q + + 2n 2 n0 t + 2q 2q + + 2q 2 q 2 + 6t 6t + 4τ 4τ

j, p = 0, j  = p

 



+(d +(dG (vi ) + dG (vk )) qn 0 + 2n 2 n0 t + 3t 3t + 2q 2q 2 + 4τ 4τ



+ 2n0 t + 2q 2q 2 + 4τ 4τ . Hence n−1

A3



=

T 3 dλG (vi , vk ) =



2H λ∗(G)

i, k = 0 i =k

  

2



2n0 q + + 2n 2 n0 t + 2q 2q + + 2q 2 q + 6t 6t + 4τ 4τ



2

+2H +2H λ(G) qn0 + 2n 2 n0 t + 3t 3t + 2q 2q + 4τ 4τ 2



+2W +2W λ (G) 2n0 t + 2q 2q + 4τ 4τ . Finally, we obtain A 4 =

i = 0 j = 0

T 4′ . By Lemma 2. 2 .4, we have T 4′

n−1 r−1

dG (Bij )dG (Bij )dλG (Bij , Bij ). For that first we calculate ′

′

′

= dG (Bij )dG (Bij ) ′

= =



′

2



− mj + 1) + (n ( n0 − mj ) 2 dG (vi )(n )(n0 − mj + 1) 2 + 2d 2dG (vi )(n )(n0 − mj )(n )(n0 − mj + 1) + (n ( n0 − mj )2 . dG (vi )(n )(n0

From Lemma 2. 2 .3, the distance between (B ( Bij and (B (Bij is m is m j (mj T 4′ mj (mj

− 1)



− 1). 1). Thus



2 = dG (vi ) mj4

− (2n (2n0 + 3)m 3) mj3 + ((n (( n0 + 1) 2 + 2)m 2)mj2 − (n0 + 1) 2 mj +2d +2dG(vi ) mj4 − (2n (2n0 + 2)m 2) mj3 + (n ( n20 + 3n 3 n0 + 1)m 1) mj2 − (n20 + n0 )mj + mj4 − (2n (2n0 + 1)m 1) mj3 + (n20 + 2n 2 n0)mj2 − n20 mj .









77


By Remark 2. 2.1, we obtain r−1

T 4

=



T 4′ mj (mj

j = 0

=



2 dG (vi )

− 1)

4n20 q

2n30



3n20

− − − 2n0t + 5n 5n0 q − 9t − 6q − n0 − 4τ +2d +2dG (vi ) 2q 2 − 2n0 t − 2q − 6t − 4τ + 2q 2 − 2n0 t − n0 q − 3t − 4τ .









Hence n−1

A4

=



T 4 dλG (Bij , Bij ) ′

i=0





= M 1 (G) 4n20 q

− 2n30 − 3n20 − 2n0t + 5n 5 n0 q − 9t − 6q − n0 − 4τ +4m +4m 2q 2 − 2n0 t − 2q − 6t − 4τ +n 2q 2 − 2n0 t − n0 q − 3t − 4τ .





 

Adding A Adding A 1 ,S 1 ,S 2 ,A3 and A and A 4 we get the required result. If we set λ = 1 in Theorem Theorem 2 .5, we obtain the product degree distance of G of G⊠K m0, m1 , ··· , mr 1 . −

Theorem 2.6 Let G G be a connected graph with n vertices and m edges. Then DD∗ (G ⊠ K m0 , m1 , ··· , mr 1 ) −

= (4q (4 q 2 + n20 + 4n 4 n0 q )DD∗ (G) + 4q 4 q 2 W ( W (G) n +(4q +(4q 2 + 2n 2n0 q )DD DD((G) + (4q (4q 2 n0 q 3t) 2 M 1(G) + 4n20 q + + 6q 6 q 2 4n0 t 15 15tt + 7n 7n0 q n0 2

−

−

  − 

3n20



2n30

− − − − − − 8τ +m n0 q 2n0 t + 2q 2q 2 − 9t − 4q − 4τ +M 2 (G) 2q 2 − 2n0 t − 3n30 + 10n 10 n0 q + + n20 − 18 18tt − 6q − n0 − 4τ





for r

≥ 2. 2 .

Setting λ = 1 in Theorem 2. 2 .5, we obtain obtain the recipr reciprocal ocal product product degree degree distance distance of G ⊠ K m0 , m1 , ··· , mr 1 .

− −

Theorem 2.7 Let G G be a connected graph with n vertices and m edges. Then RDD∗ (G ⊠ K m0, m1 , ··· , mr 1 ) −

2

= (4q (4 q +

n20 + 4n 4 n0 q )RDD∗ (G) + 4q 4 q 2 H (G)

+(4q +(4q 2 + 2n 2n0q )RDD( RDD (G) +

n (4q (4q 2 2

− n0q − 3t)

78

K.Pattabiraman

M 1 (G) 4n20 q + + 2n 2 n0 t + 3t 3t + 7n 7n0 q n0 3n20 2n30 2 5n0 q 9t 9 t +m + n0 t q 2 4q + + 2τ 2 τ 2 2 M 2 (G) 2q 2 2n0 t 3n30 + 10n 10 n0 q + + n20 18 18tt 6q 2 +

−



 

− −

− − − − −



−



− 2q + + 4τ 4 τ



− − − n0 − 4τ

for r

≥ 2. 2 .

References [1] A.R. Ashrafi, T. Doslic and A. Hamz Hamzeha, eha, The Zagreb coindices coindices of graph operations, operations, Discrete Appl. Math., 158 (2010) 1571-1578. [2] N. Alon, E. Lubetzky, Lubetzky, Independent set in tensor graph powers, J. Graph Theory , 54 (2007) 73-87. [3] Y. Alizadeh, A. Iranmanesh Iranmanesh,, T. Doslic, Additively Additively weighted weighted Harary index of some composite graphs, Discrete Math., 313 (2013) 26-34. [4] A.M. Assaf, Modified group divisible divisible designs, Ars Combin., 29 (1990) 13-20. [5] B. Bresar, W. Imrich, Imrich, S. Klavˇ Klavˇ zar, zar, B. Zmazek, Zmazek, Hypercubes Hypercubes as direct direct products, products, SIAM J. Discrete Math., 18 (2005) 778-786. [6] A.A. Dobrynin, Dobrynin, A.A. Kochetov Kochetova, Degree distance of a graph: a degree analogue of the Wiener index, J. Chem. Inf. Comput. Sci. , 34 (1994) 1082-1086. [7] S. Chen , W. Liu , Extremal Extremal modified modified Schultz Schultz index of bicyclic bicyclic graphs, MATCH Commun. Math. Math. Comput. Comput. Chem. Chem., 64(2010)767-782. [8] J. Devillers, A.T. Balaban, Eds., Topological Indices and Related Descriptors in QSAR and QSPR, Gordon and Breach, Amsterdam, The Netherlands, 1999. QSPR/QRAR Studies Studies by Molecula Molecularr Descripto Descriptors rs , Nov [9] M.V. Diudea(Ed. Diudea(Ed.), ), QSPR/QRAR Nova, a, Huntingt Huntington on (2001). [10] B. Furtula, urtula, I.Gutman I.Gutman,, Z. Tomov omovic, ic, A. Vesel, esel, I. Pesek, Pesek, Wiener-t Wiener-type ype topological topological indices indices of phenylenes, Indian J. Chem. , 41A(2002) 1767-1772. Chem. Inf. [11] I. Gutman, Selected properties properties of the Schultz Schultz molecular molecular topological topological index, J. Chem. Comput. Sci., 34 (1994) 1087-1089. [12] I. Gutman, O.E. Polansky, Polansky, Mathematical Concepts in Organic Chemistry , Springer-Verlag, Springer-Verlag, Berlin, 1986. [13] I. Gutman, A property of the Wiener number and its modifications, Indian J. Chem. , 36A(1997) 128-132. [14] I. Gutman, A.A. Dobrynin, Dobrynin, S. Klavzar. Klavzar. L. Pavlovic, Pavlovic, Wiener-t Wiener-type ype invarian invariants ts of trees and their relation, Bull. Inst. Combin. Appl. , 40(2004)23-30. [15] I. Gutman, D.Vidovic and L. Popovic, Graph Graph representation of organic organic molecules. Cayley’s araday Trans., 94 (1998) 857-860. plerograms vs. his kenograms, J.Chem. Soc. Faraday [16] A. Hamzeh, A. Iranmanesh, S. Hossein-Zadeh , M.V. Diudea, Generalized degree distance distance of trees, unicyclic and bicyclic graphs, Studia Ubb Chemia, LVII, 4(2012) 73-85. [17] A. Hamzeh, A. Iranmanesh, S. Hossein-Zadeh, Some results on generalized degree distance, Open J. Discrete Math. , 3 (2013) 143-150.


79

Discrete Appl. Math., 160 [18] H. Hua, S. Zhang, Zhang, On the reciprocal reciprocal degree distance distance of graphs, Discrete (2012) 1152-1163. Product graphs: graphs: Structure and Reco Recognition gnition , John Wiley, New York [19] W. Imrich, Imr ich, S. Klavˇzar, zar, Product (2000). [20] S.C. Li, X. Meng, Four Four edge-graftin edge-grafting g theorems on the reciprocal reciprocal degree distance of graphs and their applications, J. Comb. Optim. , 30 (2015) 468-488. [21] A. Mamut, E. Vumar, Vertex Vertex vulnerability parameters of Kronecker products of complete Inform. Process. Process. Lett., 106 (2008) 258-262. graphs, Inform. [22] K. Pattabiraman, M. Vijayaragav Vijayaragavan, an, Reciprocal degree distance of some graph operations, Trans. Comb., 2(2013) 2(2013) 13-24. [23] K. Pattabiraman, Pattabiraman, M. M. Vijayaragav Vijayaragavan, an, Reciprocal degree distance distance of product graphs, Discrete Appl. Math., 179(2014) 179(2014) 201-213. [24] K. Pattabiraman Pattabiraman,, Generalizatio Generalization n on product degree distance distance of tensor tensor product of graphs, J. Appl. Math. & Inform. , 34(2016) 341- 354. [25] K. Pattabiraman Pattabiraman,, P. Kandan, Kandan, Generaliz Generalized ed degree distance distance of strong strong product of graphs, graphs, Iran. J. Math. Sci. & Inform. , 10 (2015) 87-98. [26] K. Pattabiraman Pattabiraman,, P. Kandan, Kandan, Generalizati Generalization on of the degree distance of the tensor product of graphs, graphs, Aus. J. Comb. , 62(2015) 62(2015) 211-227. [27] G. F. Su, L.M. Xiong, X.F. Su, X.L. Chen, Some results on the reciprocal sum-degree distance of graphs, J. Comb. Optim. , 30(2015) 30(2015) 435-446. [28] G. Su, I. Gutman, Gutman, L. Xiong, L. Xu, Reciprocal product degree distance of graphs , Manuscript. Chem. m. Inf. Inf. [29] H. Y. Zhu, D.J. Klenin, I. Lukovits, Extensions of the Wiener number, J. Che Comput. Sci., 36 (1996) 420-428.


Semifull Line (Block) Signed Graphs V. Lokesha1, P. S. Hemavathi 1,2 and S. Vijay3 1. Department of Studies in Mathematics, Vijayanagara Sri Krishnadevaraya University, Ballari-583 105, India 2. Department of Mathematics, Siddaganga Institute of Technology, B.H.Road, Tumkur-572 103, India 3. Dept. of Mathematics, Mathematics, Governmen Governmentt Science College, College, Hassan-573 201, India E-mail: [email protected]

Abstract: Abstract: In this paper we introduced the new notions semifull signed graph and semifull line (block) signed graph of a signed graph and its properties are obtained. Also, we obtained the structural structural charact characteriz erizatio ations ns of these notions. notions. Further, urther, we presented presented some switch switching ing equivalent characterizations.

Key Words: Words: Signed graphs, balance, switching, semifull signed graph, semifull line (block) signed graph, negation of a signed graph, semifull Smarandachely graph.

AMS(2010): AMS(2010): 05C22. §1. Introduction

For all terminology and notation in graph theory we refer the reader to consult any one of the standard standard text-books by Chartrand Chartrand and Zhang [2], Harary [3] and West [12].

{

···

≥ }

If B B = u1 , u2, , ur , r 2 is a block of a graph Γ, then we say that vertex u 1 and block B are incident with each other, as are u 2 and B and B and so on. If two blocks B 1 and B and B 2 of G G are incident with a common cut vertex, then they are adjacent blocks. If B If B = = e1 , e2 , , es , s 1 is a block of a graph Γ, then we say that an edge e 1 and block B are incident with each other, as are e are e 2 and B and B and so on. This concept concept was introduced introduced by Kulli [7]. The vertices, vertices, edges edges and blocks of a graph are called its members.

{

···

≥ }

The line graph L graph L(Γ) (Γ) of a graph Γ is the graph whose vertex set is the set of edges of Γ in which two vertices are adjacent if the corresponding edges are adjacent (see [3]). The semifull graph ( Γ) of a graph Γ is the graph whose vertex set is the union of (Γ) vertices, edges and blocks of Γ in which two vertices are adjacent if the corresponding members of Γ are adjacent or one corresponds to a vertex and the other to an edge incident with it or one corresponds to a block B of Γ and the other to a vertex v of Γ and v is in B . In fact, this notion notion was introduced introduced by Kulli [8]. Generally Generally,, for a subset B ′ B, B , a semifull Smarandachely ′ graph (Γ) of a graph Γ on B is the graph with V ( V ( (Γ)) (Γ)) = V (Γ) V (Γ) E (Γ) (Γ) B ′ , and two vertices are adjacent in (Γ) if the corresponding members of Γ are adjacent or one corresponds to a vertex and the other to an edge incident with it or one corresponds to a block

SF S F

SSF S SF


SSF S SF


⊂ SSF SS F

 

81

Semifull Line (Block) Signed Graphs

B ′ of Γ and the other to a vertex v of Γ with v

∈ B ′. Clearly, SSF SS F (Γ) (Γ) = SF (Γ) if B B ′ = B. B .

In [9], the author introduced the new notions called “ semifull line graphs and semifull block graphs ” as follows: follows: The semifull semifull line graph (Γ) of a graph Γ is the graph whose vertex set is the union of the set of vertices, edges and blocks of Γ in which two vertices are adjacent (Γ) if the corresponding vertices and edges of Γ are adjacent or one corresponds to a vertex of Γ and other to an edge incident with it or one corresponds to a block B of Γ and other to a vertex v of Γ and v is in B in B .

SF S F L

SFL SF L

The semifull block graph (Γ) of a graph Γ is the graph whose vertex set is the union of the set of vertices, edges and blocks of Γ in which two vertices are adjacent in (Γ) if the corresponding vertices and blocks of Γ are adjacent or one corresponds to a vertex of Γ and other to an edge incident with it or one corresponds to a block B of Γ and other to a vertex v of Γ and v and v is in B in B..

SFB SF B

SF S F B

A signed graph is an ordered pair Σ = (Γ, (Γ , σ), where Γ = (V, (V, E ) is a graph called underlying graph of Σ and σ : E +, is a functi function. on. We say that that a signed signed graph graph is connected if its underlyin underlying g graph is connected connected.. A signed graph Σ = (Γ, (Γ , σ ) is balanced , if every cycle in Σ has an even number of negative edges (See [4]). Equivalently, a signed graph is balanced if product of signs of the edges on every cycle of Σ is positive.

→ { −}

∼

Signed graphs Σ1 and Σ2 are isomorphic, written Σ 1 = Σ2 , if there is an isomorphism between their underlying graphs that preserves the signs of edges. The theory of balance goes back to Heider [6] who asserted that a social system is balanced if there is no tension and that unbalanced social structures exhibit a tension resulting in a tenden tendency cy to change change in the directi direction on of balanc balance. e. Since Since this this first first work work of Heider Heider,, the notion notion of balance balance has been extensively extensively studied by many many mathemat mathematicia icians ns and psychologist psychologists. s. In 1956, Cartwright and Harary [4] provided a mathematical model for balance through graphs. A marking of Σ is a function ζ : V (Γ) V (Γ) +, . Given Given a signed graph Σ one can easily define a marking ζ marking ζ of of Σ as follows: For any vertex v V (Σ), V (Σ),

→ { −} ∈

ζ (v) =



σ(uv) uv ),

(Σ) uv∈ uv∈E (Σ)

the marking ζ marking ζ of of Σ is called canonical marking of Σ. The following are the fundamental results about balance, the second being a more advanced form of the first. Note that in a bipartition of a set, V = V 1 V 2 , the disjoint subsets may be empty.

∪

Theorem 1.1 A signed graph Σ is balanced if and only if either of the following equivalent conditions is satisfied: (1)(Harary [4]) Its vertex set has a bipartition V = V 1 V 2 such that every positive edge joins vertices vertices in V 1 or in V V 2 , and every negative edge joins a vertex in V 1 and a vertex in V 2 .

∪

(2)(Sampathkumar [10]) There exists a marking µ of its vertices such that each edge uv in Γ Γ satisfies σ(uv) uv ) = ζ ( ζ (u)ζ (v). Let Σ = (Γ, (Γ , σ ) be a signed graph. Complement of Σ is a signed graph Σ = (Γ , σ ′ ), where

82

V. Lokesha, P. S. Hemavathi Hemavathi and S. Vijay

for any edge e = uv Γ, σ′ (uv) uv) = ζ (u)ζ (v). Clearly Clearly, Σ as defined here is a balanced signed signed graph due to Theorem 1.1. A switching function for Σ is a function ζ : V +, . The switc switched hed signat signature ure is ζ ζ σ (e) := ζ ( ζ (v)σ(e)ζ (w), where e where e has has end points v, points v, w. The switched switched signed signed graph is Σ := (Σ σζ ). We say that Σ switched by ζ . ζ . Note that Σ ζ = Σ −ζ (see [1]). If X V , switching Σ by X X (or simply switching X ) means reversing the sign of every c edge in the cutset E ( E (X, X ). The switched signed graph is Σ X . This is the same as Σ ζ where ζ (v) := if and only if v X . Switc Switchin hingg by ζ or X X is the same operation with different X Xc notation. Note that Σ = Σ . Signed graphs Σ1 and Σ2 are switching equivalent, written Σ 1 Σ 2 if they have the same underlyin underlyingg graph and there exists a switching switching function function ζ such ζ such that Σζ equivalencee 1 = Σ 2 . The equivalenc class of Σ, [Σ] := Σ′ : Σ ′ Σ

∈

→ { −}

|

⊆ ⊆

−

∈

∼

{

∼

∼ }

is called the its switching switching class.

∼

Similarly, Σ 1 and Σ2 are switching isomorphic, written Σ 1 = Σ 2 , if Σ1 is isomorphic to a switching of Σ2 . The equivalence class of Σ is called its switching isomorphism class. Two signed graphs Σ 1 = (Γ1 , σ1 ) and Σ2 = (Γ2 , σ2 ) are said to be weakly isomorphic (see [11]) or cycle isomorphic (see [13]) if there exists an isomorphism φ : Γ1 Γ2 such that the sign of every cycle Z in Σ1 equals to the sign of φ( φ (Z ) in Σ2 . The following result is well known (see [13]):

→

Theorem 1 Theorem 1..2(T. Zaslavsky, [13]) Two signed graphs Σ 1 and Σ Σ 2 with the same underlying graph are switching equivalent if and only if they are cycle isomorphic.

Semifull Line Signed Graphs §2. Semifull Motivated by the existing definition of complement of a signed graph, we now extend the notion called called semifull semifull line graphs to realm of signed graphs: graphs: the semifull line signed graph (Σ) ′ of a signed graph Σ = (Γ, (Γ , σ ) as a signed graph (Σ) = ( (Γ), (Γ), σ ), where for any edge e1 e2 in (Γ), σ ′ (e1 e2 ) = σ(e1 )σ(e2 ). Further urther,, a signed signed graph graph Σ = (Γ, (Γ , σ) is called semifull ′ ′ line signed graph, if Σ = (Σ ) for some signed graph Σ . The following following result result indicates indicates the limitations of the notion of semifull line signed graphs as introduced above, since the entire class of unbalanced signed graphs is forbidden to be semifull line signed graphs.

SFL SF L

SF S F L

SF S F L

SFL SF L

∼ SF S F L

SF S F L(Σ) is

signed graph graph Σ = (Γ, semifull line signed signed graph graph Theorem 2.1 For any signed (Γ , σ ), its semifull balanced. ′

SF S F L

Proof Proof Let Let σ denote the signing of (Σ) and let the signing σ of Σ be treated as a marking of the vertices of (Σ). (Σ). Then Then by definit definition ion of (Σ), we see that σ′ (e1 e2 ) = σ(e1 )σ(e2 ), for every edge e 1 e2 of (Σ) and hence, by Theorem 1, the result follows.

SF S F L

SFL SF L

SF S F L

For any positive integer k , the k th iterated semifull line signed graph,

SF S F Lk (Σ) of Σ is

83


defined as follows:

SFL SF L0(Σ) = Σ, SFL SF Lk (Σ) = SFL SF L(SFL SF Lk−1 (Σ)) graph Σ Corollary 2.2 For any signed graph Σ = (Γ, (Γ, σ ) and for any positive integer k , balanced.

SF S F Lk(Σ) is

Proposition 2 Proposition 2..3 For any two signed graphs Σ1 and Σ2 with the same underlying graph, their semifull line signed graphs are switching equivalent.

∼

Proof Suppose Σ1 = (Γ, (Γ, σ) and Σ2 = (Γ′ , σ ′ ) be two signed graphs with Γ = Γ′ . By Theorem 2. 2.1, (Σ1 ) and (Σ2 ) are balanced and hence, the result follows from Theorem 1.2.

SFL SF L

SFL SF L

The semifull signed graph (Σ) of a signed graph Σ = (Γ, (Σ) (Γ , σ) as a signed graph (Σ) (Σ) = ′ ′ ( (Γ), (Γ), σ ), where for any edge e 1 e2 in (Γ), σ (Γ), σ (e1 e2 ) = σ( σ (e1 )σ(e2 ). Further, a signed graph Σ = (Γ, (Γ, σ ) is called semifull signed graph, if Σ = (Σ (Σ′ ) for some signed graph Σ ′ . The following result indicates the limitations of the notion of semifull signed graphs as introduced above, since the entire class of unbalanced signed graphs is forbidden to be semifull signed graphs.

SF S F

SF

SF

SF

∼ SF S F

Theorem 2.4 For any signed graph Σ = (Γ, (Γ, σ ), its semifull signed graph

SF SF (Σ) is balanced.

′

Proof Let σ Let σ denote the signing of (Σ) (Σ) and let the signing σ of σ of Σ be treated as a marking of the vertices of (Σ). Then by definition of (Σ). (Σ), (Σ), we see that σ ′ (e1 e2 ) = σ( σ (e1 )σ (e2 ), for every edge e edge e 1 e2 of (Σ) and hence, by Theorem 1, the result follows. (Σ)

SF

SF SF SF SF

SF SF

For any positive integer k, the kth iterated semifull line signed graph, defined as follows:

SF S F k (Σ) of Σ is

SF 0(Σ) = Σ, SF k (Σ) = SF (SF k−1 (Σ)) For any signed signed graph Σ = (Γ, Corollary 2.5 For (Γ , σ ) and for any positive integer k , balanced.

SF S F k(Σ) is

Proposition 2 Proposition 2..6 For any two signed graphs Σ1 and Σ2 with the same underlying graph, their semifull signed graphs are switching equivalent.

∼

Proof Suppose Σ1 = (Γ, (Γ, σ) and Σ2 = (Γ′ , σ ′ ) be two signed graphs with Γ = Γ′ . By Theorem 2. 2.4, (Σ (Σ1 ) and (Σ (Σ2 ) are balanced and hence, the result follows from Theorem 1.2.

SF S F

SF S F

In [9], the author characterizes graphs such that semifull line graphs and semifull graphs are isomorphic.

nontrivial al conne connecte cted d graph graph.. The graphs graphs Theorem 2.7 Let Γ be a nontrivi isomorphic if and only if Γ is a block.

SF S F L(Γ) and SF S F (Γ) (Γ) are

84


In view of the above result, we have the following result that characterizes the family of signed graphs satisfies (Σ) (Σ). (Σ).

SFL SF L

∼ SF

Theorem 2.8 For any signed graph Σ = (Γ, (Γ, σ ),

Proof Suppose that Theorem 2. 2.7, Γ is a block.

SF S F L(Σ) ∼ SF (Σ). (Σ).

SFL SF L(Σ) ∼ SF (Σ) if and only if Γ is a block. Then Then clearly clearly,, SFL SF L(Γ) ∼= SF (Γ). (Γ). Henc Hencee by

Conversely, suppose that Σ is a signed graph whose underlying graph is a block. Then by Theorem 2. 2.7, (Γ) and (Γ) (Γ) are isomorphic. isomorphic. Since for any signed graph Σ, both (Σ) and (Σ) are balanced, the result follows by Theorem 1 .2. (Σ)

SF

SFL SF L

SF

SFL SF L

The following result characterize signed graphs which are semifull line signed graphs. Theorem 2.9 A signed graph Σ = (Γ, (Γ, σ ) is a semifull line signed graph if and only if Σ is balanced signed graph and its underlying graph Γ is a semifull line graph.

Proof Suppose that Σ is balanced and Γ is a semifull semifull line line graph. Then there there exists a graph ′ Γ such that (Γ ) = Γ. Since Since Σ is balanced balanced,, by Theorem Theorem 1.1, there there exists exists a markin markingg ζ of Γ such that each edge uv in Σ satisfies σ(uv) uv ) = ζ (u)ζ (v). Now consider consider the signed signed graph ′ ′ ′ ′ ′ Σ = (Γ , σ ), where for any edge e in Γ , σ (e) is the marking of the corresponding vertex in Γ. Then clearly, (Σ′ ) = Σ. Hence Σ is a semifull line signed graph. ′

SF S F L ∼ SFL SF L

∼

Conversely, suppose that Σ = (Γ, (Γ , σ ) is a semif semifull ull line line signed signed graph. graph. Then Then there there exists exists a signed graph Σ′ = (Γ ′ , σ ′ ) such that

SFL SF L(Σ′) ∼= Σ. Σ. Hence, Γ is the semiful line graph of Γ ′ and by Theorem 2. 2 .1, Σ is balanced. In view of the above result, we can easily characterize signed graphs which are semifull signed graphs.

negation η (Σ) of a given signed graph Σ defined in [5] as follows: The notion of negation η(Σ) has the same underlying graph as that of Σ with the sign of each edge opposite to that given to it in Σ. Howev However, er, this this definit definitio ion n does does not say anyth anythin ingg about about what to do with with nonadjacent pairs of vertices in Σ while applying the unary operator η (.) of taking the negation of Σ. For a signed graph Σ = (Γ, (Γ , σ), the conditions under which negation η(Σ) η (Σ) of

SFL SF L(Σ) (SF (Σ)) (Σ)) is balanced. balanced. We now examine, the SFL SF L(Σ) (SF (Σ)) (Σ)) is balanced. be a signe signed d graph graph.. If SFL Theorem 2.10 Let Σ = (Γ, (Γ, σ ) be SF L(Γ) ( SF SF (Γ) (Γ)) is bipartite then η (SFL SF L(Σ)) ( η(SF (Σ)) (Σ))) is balanced. Proof Since SFL SF L(Σ) (SF (Σ)) (Σ)) is balanced, if each cycle C in SFL SF L(Σ) (SF (Σ)) (Σ)) contains even number number of negative negative edges. edges. Also, since SFL SF L(Γ) (SF (Γ)) (Γ)) is bipartite, all cycles have even length; thus, the number of positive edges on any cycle C in SF L(Σ) (SF (Σ)) (Σ)) is also even. SF L(Σ)) (η Hence η Hence η((SFL (η (SF (Σ))) is balanced.

85


§3. Semifull Semifull Block Signed Graphs

Motivated by the existing definition of complement of a signed graph, we now extend the notion called semifull block graphs to realm of signed graphs: the semifull block signed graph (Σ) ′ of a signed graph Σ = (Γ, (Γ , σ) as a signed graph (Σ) = ( (Γ), (Γ), σ ), where for any edge e1 e2 in (Γ), σ ′ (e1 e2 ) = σ(e1 )σ(e2 ). Further urther,, a signed signed graph graph Σ = (Γ, (Γ , σ) is called semifull ′ ′ block signed graph, if Σ = (Σ ) for some signed graph Σ . The following following result indicates indicates the limitations of the notion of semifull block signed graphs as introduced above, since the entire entire class of unbalanced unbalanced signed graphs is forbidden to be semifull semifull block signed graphs.

SFB SF B

SF S F B

SF S F B

SFB SF B

∼ SFL SF L

SF S F B(Σ) is

signed graph Σ = (Γ, Theorem 3.1 For any signed (Γ , σ ), its semifull block signed graph balanced. ′

Proof Let σ denote the signing of (Σ) and let the signing σ of Σ be treated as a marking of the vertices of (Σ). (Σ). Then Then by definit definition ion of (Σ), we see that σ′ (e1 e2 ) = σ(e1 )σ(e2 ), for every edge e 1 e2 of (Σ) and hence, by Theorem 1, the result follows.

SF S F B

SF S F B

SFB SF B

SF S F B

For any positive integer k , the k th iterated semifull block signed graph, defined as follows:

SFB SF Bk (Σ) of Σ is

SFB SF B0(Σ) = Σ, SFB SF Bk (Σ) = SFB SF B(SFB SF Bk−1 (Σ)) graph Σ Corollary 3.2 For any signed graph Σ = (Γ, (Γ, σ ) and for any positive integer k , balanced.

SF S F B k(Σ) is

Proposition 3 Proposition 3..3 —it For For any two two signed graphs graphs Σ 1 and Σ2 with the same underlying graph, their semifull block signed graphs are switching equivalent.

∼

Proof Suppose Σ1 = (Γ, (Γ, σ) and Σ2 = (Γ′ , σ ′ ) be two signed graphs with Γ = Γ′ . By Theorem 3. 3.1, (Σ1 ) and (Σ2 ) are balanced and hence, the result follows from Theorem 1.2.

SFB SF B

SFB SF B

In [9], the author characterizes graphs such that semifull block graphs and semifull graphs are isomorphic.

SF S F B(Γ) and SF S F (Γ) (Γ) are

nontrivial al conne connecte cted d graph graph.. The graphs graphs Theorem 3.4 Let Γ be a nontrivi isomorphic if and only if Γ is P 2 .

In view of the above result, we have the following result that characterizes the family of signed graphs satisfies (Σ) (Σ). (Σ).

SFB SF B

∼ SF

Theorem 3.5 For any signed graph Σ = (Γ, (Γ, σ ),

SFB SF B(Σ) ∼ SF (Σ) if and only if Γ is P 2. Then Then clearly clearly,, SFB SF B(Γ) ∼= SF (Γ). (Γ). Henc Hencee by

Proof Suppose that (Σ) (Σ). (Σ). Theorem 16, Γ is P 2 . Conversely, suppose that Σ is a signed graph whose underlying graph is P 2 . Then Then by Theorem 16, (Γ) and (Γ) are isomorphic. Since for any signed graph Σ, both (Γ) (Σ) and (Σ) are balanced, the result follows by Theorem 2. (Σ)

SF S F B

SF

SFB SF B

SF

∼ SF

SFB SF B

86


In view of the Theorem 2 .9, we can easily characterize signed graphs which are semifull block signed graphs. Acknowledgement The authors are thankful to the anonymous referee for valuable suggestions and comments for the improvement of the paper.

References [1] R. P. Abelson and M. J. Rosenberg, Symoblic Symoblic psychologic:A model of attitudinal cognition, Behav. Sci., 3 (1958), 1-13. [2] G.T. Chartrand Chartrand and P. Zhang, An Introduction to Graph Theory , Walter Rudin Series in Advanced Mathematics, Mc- Graw Hill Companies Inc., New York (2005). [3] F. Harary, Harary, Graph Theory , Addison-Wesley Publ. Comp., Massachusetts, Reading (1969). Michiga igan n Math. J., 2 (1953), [4] F. Harary, On the notion of balance of a signed graph, Mich 143-146. [5] F. Harary, Structural duality duality, Behav. Sci. , 2(4) (1957), 255-265. [6] F. Heider, Heider, Attitud Attitudes es and Cognitiv Cognitivee Organisatio Organisation, n, Journal of Psychology , 21 (1946), 107112. [7] V. R. Kulli, The semitotal block graph and the total block graph of a graph, Indian J. Pure and Appl.Math., 7 (1976), 625-630. [8] V.R. Kulli, Kulli, The semifull semifull graph of a graph, Annals of Pure and Applied Mathematics , 10(1) (2015), 99-104. J. Comp. Comp. & Math. Math. Sci. Sci. , [9] V. R. Kulli, Kulli, On semifull semifull line graphs and semifull semifull block graphs, J. 6(7) (2015), 388-394. Nat. Acad. ad. Sci. Sci. Letters etters , 7(3) [10] E. Sampathku Sampathkumar, mar, Point signed signed and line signed signed graphs, Nat. (1984), 91-93. [11] T. Sozánsky, Enueration of weak isomorphism classes of signed graphs, J. Graph Theory , 4(2)(1980), 127-144. [12] D.B. West, West, Introduction to Graph Theory , Prentice-Hall of India Pvt. Ltd., 1999. [13] T. Zaslavsky, Zaslavsky, Signed graphs, Discrete Appl. Math. , 4(1) (1982), 47-74.


Accurate Independent Domination in Graphs B.Basavanagoud (Department of Mathematics, Karnatak University, Dharwad-580 003, India)

Sujata Timmanaikar (Department of Mathematics, Government Engineering College, Haveri-581 110, India) E-mail: [email protected], [email protected]

set D of a graph G graph G = (V, E ) is an independent dominating set , if Abstract: Abstract: A dominating set D the induced subgraph D has no edges. edges. An independent independent dominating set set D D of of G is G is an accurate independent dominating set if V − D has no independent dominating set of cardinality | cardinality |D D|.

The accurate independent domination number ia (G) of G G is the minimum cardinality of an accurate independent dominating set of G. In this paper, paper, we initiate initiate a study study of this this new parameter and obtain some results concerning this parameter.

Key Words: Words: Domination, independent domination number, accurate independent domination number, Smarandache H Smarandache H -dominating -dominating set.


All graphs considered here are finite, nontrivial, undirected with no loops and multiple edges. For graph theoretic terminology we refer to Harary [1]. Let G Let G = = (V, E ) be a graph with V = p = p and and E = q = q . Let ∆(G ∆(G)(δ )(δ (G)) denote the maximum (minimum) degree and x ( x ) the least (greatest) integer greater(less) than or equal to x. The neighborhood of a vertex u is the set N ( N (u) consisting of all vertices v which are adjacent with u. The The closed neighborhood is N [ N [u] = N ( N (u) u . A set of verti vertices ces in G is independent if no two of them are adjacent. adjacent. The largest number number of vertices vertices in such a set is called the vertex independence number of G and is denoted by β o (G). For any set S set S of of vertices of G, G , the induced subgraph S is maximal subgraph of G of G with vertex set S set S .. The corona of two graphs G1 and G2 is the graph G = G1 G2 formed from one copy of G1 and V ( V (G1 ) copies of G2 where the ith vertex of G1 is adjacent to every vertex in the ith copy of G2 . A wounded spider is the graph formed formed by subdividing subdividing at most n 1 of the edges of a star K 1,n for n 0. Let Ω(G Ω(G) be the set of all pendant vertices of G, that G, that is the set of vertices vertices of degree 1. A vertex v is called a support vertex if v is neighbor of a pendant vertex and dG (v) > 1. 1 . Denote by X (G) the set of all support vertices in G, M ( M (G) be the set

⌈ ⌉⌊ ⌋

| |

| |

∪{ }



|

◦

|

≥

−

1 Supported by University Grant Commission(UGC), New Delhi, India through UGC-SAP- DRS-III, 20162021: F.510/3/DRS-III/2016 (SAP-I). 2 Received January 11, 2018, Accepted May 25, 2018.

88

B. Basavanagoud Basavanagoud and Sujata Timmanaikar Timmanaikar

of vertices which are adjacent to support vertex and J ( J (G) be the set of vertices which are not adjacent to a support vertex. The diameter diam( diam(G) of a connected graph G is the maximum distance between two vertices of G, of G, that is diam( diam(G) = max u,v∈ V is u,v ∈V ( V (G) dG (u, v ). A set B a 2-packing if for each pair of vertices u, v B, B , N G[u] N G [v] = φ A proper coloring of a graph G graph G = = (V (G (G), E (G)) is a function from the vertices of the graph to a set of colors such that any two two adjacent adjacent vertices vertices have different different colors. colors. The chromatic chromatic number χ(G) is the minimum minimum number number of colors colors needed needed in a proper proper colorin coloringg of a graph. graph. A dominator coloring of a graph G is a proper coloring in which each vertex of the graph dominates every every vertex vertex of some color class. class. The dominator chromatic number χd (G) is the minimum number of color classes in a dominator coloring of a graph G. This concept was introduced by R. Gera at.al [3]. A set D set D of of vertices in a graph G graph G = = (V, E ) is a dominating set of G, G, if every vertex in V D is adjacent to some vertex in D. D . The domination number γ (G) of G of G is is the minimum cardinality of a dominating set. For a comprehensive survey of domination in graphs, see [4, 5, 7]. -dominating Generally, if D H , such a dominating set D is called a Smarandache H -dominating set . A domin independent dominating dominating set , if the dominat atin ing g set D of a graph G = (V, E ) is an independent induced subgraph D has no edges, i.e., a Smarandache H -dominating -dominating set with E (H ) = . The independent domination number i(G) is the minimum cardinality of an independent dominating set. A dominating set D set D of of G = G = (V, E ) is an accurate dominating set if V V D has no dominating set of cardinality D . The accurate domination number γ a (G) of G is G is the minimum cardinality of an accurate accurate dominating dominating set. This concept was introduced introduced by Kulli Kulli and Kattimani Kattimani [6, 9]. An independent dominating set D set D of of G is G is an accurate independent dominating set if V V D has no independent dominating set of cardinality D . The The accurate independent domination number ia (G) of G is the minimum cardinality of an accurate independent dominating set of G. This concept was introduced by Kulli [8]. For example, we consider the graph G in Figure 1. The accurate independent dominating sets are 1, 2, 6, 7 and 1, 3, 6, 7 . Therefore i Therefore i a (G) = 4.

∈

⊆

∩

− −

   ≃  

∅

−

| |

− −

| |

{

}

{

}

2

1

6

3

4

5

7

G :

Figure 1

§2. Results

Observation 2.1 1. Every accurate independent dominating set is independent and dominating. Hence it is a minimal dominating set.

89

Accurate Independent Domination in Graphs

2. Every Every minimal minimal accurate independen independentt dominatin dominatingg set is a maximal maximal independent independent dominating set.

connected graph graph G, γ (G) Proposition 2 Proposition 2..1 For any nontrivial connected

≤ ia(G).

Proof Clearly, Clearly, every accurate independent dominating set of G of G is a dominating set of G of G.. Thus result holds. Proposition 2.2 If G contains an isolated vertex, then every accurate dominating set is an accurate independent dominating set. Now we obtain the exact values of i of i a (G) for some standard class of graphs. Proposition 2 Proposition 2..3 For graphs P p , W p and K K m,n m,n , there are

⌈ ⌉

(1) i (1) i a (P p ) = p/3 p/3 if p

≥ 3;

≥ 5; (3) i (3) i a (K m,n m,n ) = m f or 1 ≤ m < n. (2) i (2) i a (W p ) = 1 if p

Theorem 2.1 For any graph G, ia(G)

≤ p − γ (G).

Proof Let D be a minimal dominating set of G. Then Then there there exist at least one accurate accurate independent dominating set in (V ( V D) and by proposition 2.1,

− −

ia (G)

≤ |V | − |D| ≤ p − γ (G).

Notice that the path P 4 achieves this bound. Theorem 2.2 For any graph G,

⌈ p/ △ +1⌉ ≤ ia(G) ≤ ⌊ p △ / △ +1⌋ and these bounds are sharp. Proof It is known known that that p/ bound holds. By Theorem 2.1,

△

+1

≤ γ (G) and by proposition 2.1, we see that the lower

ia(G)

Notice that the path P p p , p

≤ p − γ (G), ≤ p − p/ △ +1 ≤ p △ / △ +1. +1.

≥ 3 achiev achieves es the lower lower bound. This completes completes the proof.

Proposition 2 Proposition 2..4 If G = K = K m1 ,m2 ,m3 ,··· ,mr , r

≥ 3, then

ia (G) = m 1 if m1 < m2 < m3

· · · < mr.

90


Theorem 2 Theorem 2..3 For any graph G G without isolated vertices γ a (G) i a (G) if G G = K m1 ,m2 ,m3 ,··· ,mr , r 3. Furthermore, the equality holds if G = P = P p ( p = 4, 4 , p 3), 3) , W p( p 5) or K m,n or 1 m < n . m,n f or 1



≤

≥



≥

≤

≤ γ a(G) and by Proposition 2.1,γ 2.1, γ a (G) ≤ i a (G). Let γ a (G) ≤ ia (G). If G = K m ,m ,m ,··· ,m , r ≥ 3 then by Proposition 2.4, ia (G) = m1 if m1 < m2 < m3 · · · < mr and also accurate domination number is ⌊ p/2 p/2⌋ + 1 i.e., γ a (G) = ⌊ p/2 p/2⌋ + 1 > 1 > m1 = i = i a (G), a contradiction. Proof Since we have γ have γ ((G)

1

2

3

r

Corollary 2.1 For any graph G, ia (G) = γ ( γ (G) if diam( diam(G) = 2. Proposition 2.5 For any graph G without isolated vertices i(G) ia (G). Furthermore, the equality holds if G = P = P p ( p ( p 3), 3) , W p ( p ( p 5) or 5) or K m,n or 1 m < n . m,n f or 1

≥

≥

≤

≤

Proof Every accurate independent independent dominating set is a independent dominating dominating set. Thus result holds. Definition 2.1 The double star S n,m n,m is the graph obtained by joining the centers of two stars K 1,n and K 1,m with an edge. Proposition 2.6 For any graph G G, i a (G)

≤ β o(G). Furthermore, the equality holds if G = G = S S n,m n,m .

Proof Since every minimal accurate independent independent dominating set is an maximal independent dominating set. Thus result holds. Theorem 2.4 For any graph G, i a (G)

≤ p − α0(G). − S Proof Let S Let S be be a vertex cover of G. of G. Then V − S is an accurate independent dominating set. Then i Then i a(G) ≤ |V − − S | ≤ p − α0(G). ≤ p − β 0(G) + 2.2.

Corollary 2.2 Fr any graph G, ia (G)

Theorem 2.5 If G is any nontrivial connected graph containing exactly one vertex of degree △ (G) = p 1, then γ (G) = i a (G) = 1.

−

Proof Let G Let G be any nontrivial connected graph containing exactly one vertex v of degree deg( deg (v) = p 1. Let D be a minimal dominating set of G of G containing vertex of degree deg( deg (v) = p = p = 1. Then D Then D is a minimum dominating set of G of G i.e.,

−

|D| = γ = γ (G) = 1. 1.

(1)

− D has no dominating set of same cardinality |D|. Therefore,

Also V Also V

|D| = i = i a (G). Hence, by (1) and (2) γ ( γ (G) = i a (G) = 1. 1.

(2)

≥

91


connecte cted d graph graph with p vertices vertices then ia (G) = p/2 Theorem 2.6 If G is a conne p/2 if and only if G = H = H K 1 ,where H is any nontrivial connected graph.

◦ ◦

| |

Proof Let D be any minimal accurate independent dominating set with D = p/2 p/2. If G = H K 1 then there exist at least one vertex v i V ( V (G) which is neither a pendant vertex nor a support vertex. vertex. Then there exist a minimal minimal accurate accurate independent independent dominating dominating set D′ containing v i such that

 ◦ ◦

∈

|D′ | ≤ |D| − {vi} ≤ p/2 p/ 2 − {vi } ≤ p/2 p/ 2 − 1, which is a contradiction to minimality of D of D..

◦ ◦

| | ⊆

Conversely, let l be the set of all pendant vertices in G = H K 1 such that l = p/2 p/2. If G = H K 1 , then there exist a minimal accurate independent dominating set D V ( V (G) containing all pendant vertices of G. of G. Hence D = l = p/ = p/2. 2.

◦ ◦

| | | |

Now we characterize the trees for which i a (T ) T ) = p Theorem 2.7 For any tree T , ia(T ) T ) = p T = K 1 , K 1,1 .

 

− ∆(T ∆(T )).

− ∆(T ∆(T )) if and only if T is a wounded spider and

Proof Suppose T Suppose T is is wounded spider. Then it is easy to verify that ia (T ) T ) = p = p ∆(T ∆(T )).

−

−

Conversely, suppose T T is a tree with ia(T ) T ) = p ∆(T ∆(T )). Let v be a vertex of maximum degree ∆(T ∆(T )) and u and u be a vertex in N ( N (v) which has degree 1. If T If T N [ N [v] = φ then φ then T T is is the star K 1,n , n 2. 2 . Thus T T is a double wounded wounded spider. spider. Assume Assume now there is at least one vertex in T N [ N [v]. Let S be S be a maximal independent set of T N [ N [v] . Then either S either S v or S u is an accurate independent dominating set of T of T . Thus p Thus p = = i i a (T ) T ) + ∆ (T (T )) S + 1 + ∆ (T (T )) p. This implies that V N ( N (v) is an accurate independent dominating set. Furthermore, N ( N (v) is also an accurate independent dominating set.

− −

≥

− −

 − −



∪ ∪ { } ≤| |

−

∪ ∪ { } ≤

− −

The connectivity of T implies T implies that each vertex in V N [ N [v] must be adjacent to at least one vertex in N ( N (v). Moreover if any vertex in V N [ N [v] is adjacent to two or more vertices in N ( N (v), then a cycle is formed. Hence each vertex in V N [ N [v ] is adjacent to exactly one vertex in N ( N (v). To show that ∆(T ∆(T )) + 1 vertices are necessary to dominate T , there must be at least one vertex in N ( N (v) which are not adjacent to any vertex in V N [ N [v] and each vertex in N ( N (v) has either 0 or 1 neighbors in V N [ N [v]. Thus T Thus T is is a wounded spider.

− −

−

−

Proposition 2 Proposition 2..7 If G is a path P P p p , p

−

≥ 3 then γ (P p) = ia(P p ).

We characterize the class of trees with equal domination and accurate independent domination nation number number in the next section. section.

§3. Characterization of (γ, ia )-Trees

For any graph theoretical parameter λ and µ, we define G to be (λ, (λ, µ)-graph if λ(G) =

92


µ(G). Here we provide a constructive characterization of ( γ, ia )-trees. To characterize (γ, ( γ, ia )-trees we introduce family τ 1 of trees T = T k that can be obtained as follows follows.. If k is a positiv positivee integ integer, er, then T k+1 can be obtained recursively from T k by the following operation. Operation O Attach a path P 3 (x,y,z) and an edge mx, mx , where m where m is a support vertex of a tree T. τ = T /obtained from P from P 5 by finite sequence of operations of O of O

{

}

Tree Tree T belongi belonging ng to famil family y τ 1

Observation 3.1 If T T τ, τ , then

∈ ∈ 1. ia(T ) T ) = ⌈ p + 1/ 1/3⌉;

2. X (T ) T ) is a minimal dominating set as well as a minimal accurate independent dominating set of T ; T ; 3. V D is totally disconnected.

 − − 

Corollary 3.1 If tree T with p X (T ) T ) .

|

≥ 5 belongs to the family τ then γ (T ) T ) = |X (T ) T )| and ia (T ) T ) =

|

Lemma 3.1 If a tree T belongs to the family τ then T is a (γ, ia )

− tree. Proof If T T = P p T is a (γ, ( γ, ia ) − tree. Now tree. Now if T T = P p p , p ≥ 3 then from proposition 2.7 T is p , p ≥ 3

then we proceed by induction induction on the number of operations operations n n((T ) T ) required required to construct construct the tree T . T . If n( n (T ) T ) = 0 then T P 5 by proposition 2.7 T is T is a (γ, (γ, ia )-tree. Assume now that T is T is a tree belonging to the family τ τ with n(T ) T ) = k, k , for some positive integer k and each tree T ′ τ ( γ, ia )-tree in which τ with n(T ′ ) < k and with V ( V (T ′ ) 5 is a (γ, ′ ′ X (T ) is a minimal accurate independent dominating set of T of T . Then T Then T can can be obtained from ′ ′ ′ a tree T belonging to τ τ by operation O where m V ( V (T ) (M ( M (T ) Ω(T Ω(T ′ )) and we add

∈

∈

≥

∈

−

−

93


path (x,y,z (x,y,z)) and the edge mx. mx. Then Then z is a pendant vertex in T and y is a support vertex and x M ( M (T ) T ). Thus S (T ) T ) = X ( X (T ′ ) y is a minimal accurate independent dominating set of T . Therefore ia (T ) T ) X (T ) T ) = X (T ′) + 1. Hence we conclude that ia (T ) T ) = ia (T ′ ) + 1. By the induction hypothesis and by observation 3.1(2) ia (T ′ ) = γ ( γ (T ′ ) = X (T ′) . In this way ia (T ) T ) = X (T ) T ) and in particular i a (T ) T ) = γ (T ) T ).

∈

∪{ } | | |

≥ |

|

|

|

|

Lemma 3.2 If T T is a (γ, ( γ, ia )

− tree, then T belongs to the family τ . If T is T is a path P p T is a (γ, (γ, ia ) − tree. It tree. It is easy to p , p ≥ 3 then by proposition 2.7 T

Proof verify that the statement is true for all trees T with T with diameter less than or equal to 4. Hence we may assume that diam that diam((T ) T ) 4. 4 . Let T be T be rooted at a support vertex m of m of a longest path P. path P. Let P be a m z path and let y let y be the neighbor of z z . Further, let x let x be a vertex belongs to M ( M (T ) T ). Let T Let T be be a (γ, ( γ, ia )-tree. Now we proceed by induction on number of vertices V ( V (T ) T ) of a (γ, ( γ, ia )tree. tree. Let T T be a (γ, (γ, ia )-tree and assume that the result holds good for all trees on V ( V (T ) T ) 1 vertices. By observation 3.1(2) since T is (γ, ia )-tree it contains minimal accurate independent dominating set D set D that contains all support vertices of a tree. In particular m, y D and D and the vertices x and z are independent in V D .

≥

−

|

|

{

}⊂

−

 − −  Let T ′ = T − − (x,y,z ( x,y,z)). Then D − {y }is dominating set of T ′ and so γ (T ′ ) ≤ γ (T ) T ) − 1. 1 .

Any dominating set can be extended to a minimal accurate independent dominating set of T by T by adding to it the vertices ( x,y,z) x,y,z) and so ia (T ) T ) ia (T ′) + 1. 1 . Hence, ia (T ′ ) γ (T ′) γ (T ) T ) + 1 ia (T ) T ) 1 i a (T ′ ). Consequently, we must have equality throughout this inequality chain chain.. In particu particular lar ia (T ′ ) = γ (T ′) and ia (T ) T ) = ia (T ′ ) + 1. By inductive hypothesis any minimal accurate independent dominating set of a tree T ′ can be extended to minimal accurate independen independentt dominatin dominatingg set of a tree T tree T by operation O operation O.. Thus T Thus T τ. τ .

≤

− ≤

≤

≤

≤

∈ ∈

As an immediate consequence of lemmas 3 .1 and 3. 3.2, we have the following characterization of trees with equal domination and accurate independent domination number. Theorem 3.1 Let T T be a tree. Then ia (T ) T ) = γ (T ) T ) if and only if T τ. τ .

∈

Accurate Independent Independent Domination Domination of Some Graph Families Families §4. Accurate In this section accurate independent domination of fan graph ,double fan graph , helm graph and gear graph are considered. We also obtain the corresponding relation between other dominating parameters and dominator coloring of the above graph families. Definition 4 Definition 4..1 A fan graph, denoted by F n can be constructed by joining n copies of the cycle graph C 3 with a common vertex. Observation 4.1 Let Let F n be a fan. Then, 1. F n is a planar undirected graph with 2 n + 1 vertices and 3n 3 n edges; 2. F n has exactly one vertex with ∆(F ∆( F n ) = p = p 3. Diam( Diam(F n ) = 2 .

− 1;

94


Theorem 4.1([2]) For a fan graph F n , n

≥ 2, 2 , χd (F n ) = 3. Proposition 4 Proposition 4..1 For a fan graph F n , n ≥ 2, 2 , ia (F n ) = 1. Proof By By Observation 4.1(2) and Theorem 2.5 result holds. Proposition 4 Proposition 4..2 For a fan graph F n , n

≥ 2,

ia (F n ) < χd(F n ).

Proof By Proposition Proposition 4.1 and Theorem 4.1, we know that that χ d (F n ) = 3. This implies that ia (F n ) < χd (F n ). Definition 4.2 A double fan graph, denoted by F 2,n isomorphic to P n + 2K 2 K 1 . Observation 4.2 1. F 2,n is a planar undirected graph with ( n + 2) vertices and (3n (3 n 2. Diam( Diam(G) = 2.

− 1) edges;

Theorem 4.2([2]) For a double fan graph F 2,n , n

double fan graph F 2,n , Theorem 4.3 For a double and i i a (F 2,n ) = 2 if n 7 .

≥

≥ 2, 2 , χd (F 2,n ) = 3. n ≥ 2, 2 , ia (F 2,2 ) = 2, ia (F 2,3 ) = 1, ia (F 2,5 ) = 3

Proof Our proof is divided into cases following. Case 1. If n = n = 2 and n 7, 7 , then F then F 2,n , n set D set D of D = 2. Hence, i Hence, i a (F 2,n ) = 2. 2.

≥

| | |

≥ 2 has only one accurate independent dominating

Case 2. If n n = 3, then thenF F 2,3 has exactly one vertex of ∆(G ∆( G) = p ia (F 2,n ) = 1 .

− 1. Then by Theorem 2.5,

Case 3. If n=5 n=5 and D and D be a independent dominating set of G of G with D = 2, then (V ( V D) also has an independen independentt dominating dominating set of cardinalit cardinality y 2. Hence D Hence D is not accurate. Let D1 be a independent dominating set with D1 = 3, then V then V D1 has no independent dominating set of cardinality 3. Then D 1 is accurate. Hence, i a (F 2,n ) = 3.

| |

| |

− −

−

Case 4. If n=4 and 6, there does not exist accurate independe independent nt dominating dominating set. Proposition 4 Proposition 4..3 For a double fan graph F 2,n , n

≥ 7,

γ (F 2,n ) = i( i (F 2,n ) = γ a (F 2,n ) = i a (F 2,n ) = 2

. Proof Let F Let F 2,n , n 7 be a Double fan graph. Then 2 k1 forms a minimal dominating set of F 2,n such that γ that γ (F 2,n ) = 2. Since this dominating set is independent and in ( V D) there is no independent dominating set of cardinality 2 it is both independent and accurate independent dominating set. Also it is accurate dominating set. Hence,

≥

− −

γ (F 2,n ) = i( i (F 2,n ) = γ a (F 2,n ) = i a (F 2,n ) = 2.


Proposition 4 Proposition 4..4 For Double fan graph F 2,n , n

95

≥ 7

ia (F 2,n )

≤ χd (F 2,n).

Proof The proof follows follows by Theorems Theorems 4. 4 .2 and 4. 4.3. For n Definition 4.3([1]) For n 4 ,the wheel W n is defined to be the graph W n = C n−1 + K 1 . Also it is defined as W 1,n = C n + K 1 .

≥

Definition 4.4 A helm H n is the graph obtained from W 1,n by attaching a pendant edge at each vertex of the n-cycle. Observation 4 Observation 4..3 A hel helm m H n is a planar planar undirected undirected graph with (2n+1) (2n+1) vertices vertices and 3n edges. Theorem 4.4([2]) For Helm graph H n , n

≥ 3, 3 , χd(H n ) = n + 1. 1. Proposition 4 Proposition 4..5 For a helm graph H n , n ≥ 3 , i a (H n ) = n. Proof Let H Let H n , n ≥ 3 be a helm graph. Then there exist a minimal independent dominating − D) has no independen set D set D with |D| = n = n and and (V (V − independentt dominatin dominatingg set of cardinali cardinality ty n. Hence D Hence D

is accurate. Therefore Therefore i i a (H n ) = n.

Proposition 4 Proposition 4..6 For a helm graph H n , n

≥ 3

γ (H n ) = i( i (H n ) = γ a (H n ) = i a (H n ) = n.

Proposition 4 Proposition 4..7 For a helm graph H n , n

≥ 3

ia (H n ) = χd (H n )

− 1.

Proof Applying Applying Proposition Proposition 4.5 , i a (H n ) = n = n = n n + + 1 χd (H n ) = n = n + 1. 1. Hence the proof.

− 1 = χd (H n ) − 1 by Theorem 4.4.4,

Definition 4 Definition 4..5 A gear graph Gn also known as a bipartite wheel graph, is a wheel graph W 1,n with a vertex added between each pair of adjacent vertices of the outer cycle. Observation 4.4 A gear graph Gn is a planar undirected graph with 2n + 1 vertices and 3n edges. Theorem 4.5([2]) For a gear graph Gn , n

≥ 3,

χd (Gn ) = 2n/3 n/3 + 2. 2.

⌈

⌉

96


Theorem 4.6 For a gear graph G n , n

≥ 3

, ia (Gn ) = n.

Proof It is clear from the definitio definition n of gear graph G n is obtained from wheel graph W 1,n with a vertex added between each pair of adjacent vertices of the outer cycle of wheel graph W 1,n . These n vertices vertices forms forms an independen independentt dominatin dominating g set in G n such that (V (V D) has no independent dominating set of cardinality n. Therefore, the set D with D with cardinality n is accurate independent dominating set of G of G n . Therefore i Therefore i a (Gn ) = n.

− −

Corollary 4.1 For any gear graph Gn , n

≥ 3, γ (Gn) = i( i(Gn ) = n − 1. Proposition 4 Proposition 4..8 For a gear graph Gn , n ≥ 3 , ia (Gn ) = γ a (Gn ).

Proposition 4 Proposition 4..9 For a graph Gn , n

≥ 3

ia (Gn ) = γ (Gn) + 1 = i = i((Gn ) + 1. 1.

Proof Applying Applying Theorem 4.6 and Corollary 4.1, we know that i a (Gn ) = n = n = n n γ (Gn ) + 1 = i = i((Gn ) + 1.

−1+1=

References [1] F. Harary, Harary, Graph Theory , Addison-Wesley, Reading, Mass, (1969). [2] K.Kavith K.Kavithaa and N.G.David, N.G.David, Dominator Dominator coloring of some classes of graphs, graphs, International Journal of Mathematical Archive , 3(11), 3(11), 2012, 3954-39 3954-3957. 57. [3] R.Gera, R.Gera, S. Horton, Horton, C. Rasmussen Rasmussen,, Dominator Dominator Colorings Colorings and Safe Clique Partitions, Partitions, Congressus Numerantium Numerantium , , (2006). [4] T. W. Haynes, S. T. Hedetniemi Hedetniemi and P. J. Slater, Slater, Fundamentals of Domination in Graphs , Marcel Dekker, Inc., New York, (1998). [5] T. W. Haynes, S. T. Hedetniemi Hedetniemi and P. J. Slater, Slater, Domination in Graphs- Advanced Topics , Marcel Dekker, Inc., New York, (1998). [6] V. R. Kulli Kulli and M.B.Katt M.B.Kattiman imani, i, The Accurate Domination Number of a Graph , Technical Report 2000:01, Dept.Mathematics, Gulbarga University, Gulbarga, India (2000). [7] V. R. Kulli, Kulli, Theory of Domination in Graphs , Vishwa International Publications, Gulbarga, India (2010). [8] V. R. Kulli, Advances in Domination Theory-I , Vishwa International Publications, Gulbarga, India (2012). [9] V. R. Kulli Kulli and M.B.Kattim M.B.Kattimani, ani, Accurate Domination Domination in Graphs, Graphs, In V.R.Kulli, V.R.Kulli, ed., Advances in Domination Theory-I , Vishwa International Publications, Gulbarga, India (2012) 1-8.


On r-Dynamic Coloring of the Triple Star Graph Families T.Deepa and M. Venkatachalam (Department (Department of Mathematics, Mathematics, Kongunadu Kongunadu Arts and Science College, Coimbatore Coimbatore - 641 029,Tamilna 029,Tamilnadu, du, India) E-mail: [email protected], [email protected] [email protected]

Abstract: Abstract: An r-dynamic coloring of a graph G is a proper coloring c of the vertices such that |c(N (v))| ))| ≥ min {r, d(v)}, for each v ∈ V ( V (G). The The r-dynamic chromatic number of a graph G graph G is is the minimum k minimum k such such that G that G has has an r an r-dynamic -dynamic coloring with k with k colors. colors. In this paper we investigate the r-dynamic chromatic number of the central graph, middle graph, total graph and line graph of the triple star graph K graph K 1,n,n,n denoted by C by C ((K 1,n,n,n ), M ), M ((K 1,n,n,n ), T ( T (K 1,n,n,n ) and L and L((K 1,n,n,n ) respectively.

Key Words: ords: Smarandachely r-dynamic coloring, r-dynamic coloring, triple star graph, central graph, middle graph, total graph and line graph.


Graphs Graphs in this paper are simple simple and finite. finite. For undefined terminologi terminologies es and notations notations see [5, 17]. Thus for a graph G, δ (G), ∆(G ∆(G) and χ and χ((G) denote the minimum degree, maximum degree and chromatic number of G respective respectively ly.. When the context is clear we write, δ, ∆ and χ for brevity. For v For v V ( V (G), let N let N ((v) denote the set of vertices adjacent to v in v in G and G and d d((v) = N ( N (v) . The r The r-dynamic -dynamic chromatic number was first introduced by Montgomery [14].

∈

|

|

An r-dynamic coloring of a graph G is a map c from V from V ((G) to the set of colors such that (i) if uv uv E (G), then c then c((u) = c( c(v) and (ii) for each vertex v V ( V (G), c(N ( N (v)) min r, d(v) , where N where N ((v) denotes the set of vertices adjacent to v, v , d( d (v) its degree and r and r is a positive integer. ′ Generally Generally,, for a subgraph subgraph G G and a coloring c on G if c(N ( N (v)) min r, d(v) for v V ( V (G G ′ ) but c(N ( N (v)) min r, d(v) for u V ( V (G′ ), such a r coloring is called a Smarandachely r-dynamic coloring coloring on G. Clea Clearl rly y, if G′ = , a Smarandach Smarandachely ely r-dynamic coloring is nothing else but the r-dynamic r -dynamic coloring.

∈

∈

\



|

∈

≺ |≤

{

}

∈

|

| |

|≥

|≥

{

{

}

}

∅

The first condition characterizes proper colorings, the adjacency condition and second condition is double-adjacency double-adjacency condition. The r-dynamic r -dynamic chromatic number of a graph G, G , written χr (G), is the minimum k such k such that G that G has has an r an r-dynamic -dynamic proper k proper k-coloring. -coloring. The 1-dynamic chromatic number of a graph G is equal to its chromatic number. The 2-dynamic chromatic number of a graph has been studied under the name dynamic chromatic number denoted by χd (G) [1-4, 8]. By simple observation, we can show that χ r (G) χ r +1 (G), however χ however χ r+1 (G) χr (G) can

≤

1 Received


−

98

T.Deepa and M. Venkatac Venkatachalam halam

be arbitraril arbitrarily y large, large, for example example χ( χ (Petersen) Petersen) = 2, χ 2, χ d (Petersen) Petersen) = 3, but χ but χ 3 (Petersen) Petersen) = 10. Thus, finding an exact values of χ χ r (G) is not trivially easy. There are many upper bounds and lower bounds for χ d (G) in terms of graph parameters. For example, for a graph G with ∆(G ∆(G) 3, Lai et al. [8] prove proved d that χd (G) ∆(G ∆(G) + 1. An upper bound for the dynamic chromatic number of a d-regular graph G in terms of χ(G) and the independence number of G, α(G), was introduced introduced in [7]. In fact, it was proved proved that χd (G) χ( χ(G) + 2log2 α(G) + 3. Taherkhani aherkhani gave gave in [15] an upper b ound for χ for χ 2 (G) in terms of the chromatic number, the maximum degree ∆ and the minimum degree δ . i.e., χ i.e., χ2 (G) χ(G) 2 (∆e (∆e)/δlog 2e ∆ + 1 .

≥

≤

≤



−

  

≤

Li et al. proved in [10] that the computational complexity of χ of χd (G) for a 3-regular graph graph is an NP-complete problem. Furthermore, Li and Zhou [9] showed that to determine whether there exists a 3-dynamic coloring, for a claw free graph with the maximum degree 3, is NP-complete.

N.Mohanapriya N.Mohanapr iya et al. [11, 12] studied studied the dynamic dynamic chromatic chromatic number number for various various graph families families.. Also, it was proven in [13] that the r dynamic chromatic number of line graph of a helm graph H graph H n is

−

 −     n

n + 1, 1,

χr (L(H n)) =

≤ ≤ ≤ n − 2, r = n = n − 1,

1, δ r

n + 2, 2, r = n and n n + 3, 3, n + 4, 4, n + 5, 5,

≡ 1 mod 3, ≡ 1 mod 3, r = n and n  r = n = n + 1 = ∆, ∆ , n ≥ 6 and 2n − 2 ≡ 0 ≡ 0 r = n = n + 1 = ∆, ∆ , n ≥ 6 and 2n − 2 

mod 5, mod 5.

In this paper, we study χr (G), the r- dynamic chromatic number of the middle, central, total and line graphs of the triple star graphs are discussed.

§2. Preliminaries

Let G Let G be be a graph with vertex set V set V ((G) and edge set E set E ((G). The middle graph [6] of G of G,, denoted by M by M ((G) is defined defined as follows. The vertex vertex set of M of M ((G) is V ( V (G) E (G). Two vertic vertices es x, x, y of M ( M (G) are adjacent in M ( M (G) in case one of the followi following ng holds: (i) x, y are in E (G) and x, y are adjacent in G. G . (ii) x (ii) x is in V in V ((G), y ), y is in E ( E (G), and x, and x, y are incident in G. G .

∪

The central graph [16] C (G) of a graph G is obtained from G by adding an extra vertex on each edge of G, and then joining each pair of vertices of the original graph which were previously non-adjacent. Let G be a graph with vertex set V ( V (G) and edge set E ( E (G). The total total graph graph [6, 16] of G, of G, denoted by T by T ((G) is defined in the following way. The vertex set of T of T ((G) is V is V ((G) E (G). Two vertices x, y of T ( T (G) are adjacent in T ( T (G) in case case one of the follo followin wingg holds: holds: (i) (i) x, y are in V ( V (G) and x and x is adjacent to y to y in G in G.. (ii) x, (ii) x, y are in E in E ((G) and x, and x, y are adjacent in G. G . (iii) x is in V ( V (G), y is in E in E ((G), and a nd x, x, y are incident in G. G .

∪

The line graph [13] of G denoted by L(G) is the graph with vertices are the edges of G

99

On r On r -Dynamic Coloring of the Triple Star Graph Families

with two vertices of L of L((G) adjacent whenever the corresponding edges of G of G are adjacent. Theorem 2.1 For any triple star graph K 1,n,n,n , the r r -dynamic chromatic number

 

2n + 1, 1, r = 1

χr (C (K 1,n,n,n )) =

3n + 1, 1, 2 4n + 1, 1,

≤ r ≤ ∆ − 1 r ≥ ∆

Proof First we apply the definition definition of central central graph on K 1,n,n,n . Let the the edge edge vv i , vi wi and wi ui be subdivided by the vertices ei (1 i n), ei′ (1 i n) and ei′′ (1 i n) in K 1,n,n,n . Clearly V Clearly V ((C (K 1,n,n,n )) = v vi : 1 i n wi : 1 i n ui : 1 i n ′ ′′ ei : 1 i n ei : 1 i n ei : 1 i n . The vertice verticess v i (1 i n) n ) induce a clique clique of order n (say K (say K n ) and the vertices v vertices v,, ui (1 i n) n ) induce a clique of order n + n + 1 (say K n+1 ) in C in C ((K 1,n,n,n ) respectively. Thus, we have χ r (C (K 1,n,n,n )) n + 1.

≤ ≤

 {



≤ ≤ } {

Case 1.



≤ ≤



{ } { ≤ ≤ } { ≤ ≤ } { ≤ ≤ } ≤ ≤

≤ ≤



≤ ≤ } { ≤ ≤ } ≤ ≤ ≥

r = 1.

{

···

}

{

···

}

{

···

}

Consider the color class C 1 = c1 , c2 , c3 , , c(2n the r-dynamic -dynamic coloring to (2n+1) and assign the r C (K 1,n,n,n ) by Algorithm 2.1.1. Thus, an easy check shows that the r adjacency condition is fulfilled. Hence, χ r (C (K 1,n,n,n )) = 2n 2n + 1. Case 2.

2

−

≤ r ≤ ∆ − 1.

Consider the color class C 2 = c1 , c2 , c3 , , c(3n the r-dynamic -dynamic coloring to (3n+1) and assign the r C (K 1,n,n,n ) by Algorithm 2.1.2. Thus, an easy check shows that the r adjacency condition is fulfilled. Hence, χ r (C (K 1,n,n,n )) = 3n 3n + 1. Case 3.

r

−

≥ ∆.

Consider the color class C 3 = c1 , c2 , c3 , , c(4n the r-dynamic -dynamic coloring to (4n+1) and assign the r C (K 1,n,n,n ) by Algorithm 2.1.3. Thus, an easy check shows that the r adjacency condition is fulfilled. Hence χ r (C (K 1,n,n,n )) = 4n 4 n + 1.

−

Algorithm 2.1.1 Input: The Input: The number ”n ”n” of K K 1,n,n,n . Output: Assigning Output: Assigning r r-dynamic -dynamic coloring for the vertices in C ( C (K 1,n,n,n ). begin for i for i = 1 to n to n

{

{ }

V 1 = ei ; C (ei ) = i; i ;

}

{}

V 2 = v ; C (v) = n + 1;

100


for i for i = 1 to n to n

{

V 3 = vi ; C (vi ) = n + n + i + 1;

{ }

}


{

V 4 = ei′ ; C (ei′ ) = n + 1;

{ }

}


{

{ }

V 5 = wi ; C (wi ) = i; i ;

}


{

V 6 = ei′′ ; C (ei′′ ) = n + 1;

{ }

}


{

V 7 = ui ; C (ui ) = i; i ;

}

{ }

V = V 1 end

      V 2

V 3

V 4

V 5

V 6

V 7 ;


{

{ }

V 1 = ui ; C (ui ) = i; i ;

}


{

V 2 = ei′′ ; C (ei′′ ) = n + 1;

}

{ }

On r On r-Dynamic -Dynamic Coloring of the Triple Star Graph Families


{

V 3 = wi ; C (wi ) = n + i + 1;

{ }

}


{

V 4 = ei′ ; C (ei′ ) = i; i ;

{ }

}


{

{ }

V 5 = vi ; C (vi ) = 2n + i + 1;

}


{

−1

{ }

V 6 = ei ; C (ei ) = 2 n + i + 2;

}

C (en ) = 2 n + 2; V 7 = v ; C (v) = n + 1; V = V 1 V 2 V 3 end

{}

      V 4

V 5

V 6

V 7 ;


{

V 1 = ui ; C (ui ) = i; i ;

}

{ } {}

V 2 = v ; C (v) = n + 1; for i for i = 1 to n to n

{

V 3 = wi ; C (wi ) = n + i + 1;

}

{ }

101

102



{

{ }

V 4 = vi ; C (vi ) = 2n + i + 1;

}


{

V 5 = ei ;

{ }

C (ei ) = 3 n + i + 1;

}


{

V 6 = ei′ ;

{ }

C (ei′ ) = i; i ;

}


{

V 7 = ei′′ ;

{ }

C (ei′′ ) = 3n 3 n + 2;

}

V = V 1 end

      V 2

V 3

V 4

V 5

V 6

V 7 ;

Theorem 2.2 For any triple star graph K 1,n,n,n , the r r -dynamic chromatic number

  

n + 1, 1, 1

χr (M ( M (K 1,n,n,n )) =

≤ r ≤ n

n + 2, 2 , r = n = n + 1 n + 3, 3, r

≥ ∆

Proof By definition definition of middle graph, each each edge vv v vi , vi wi and wi ui be subdivided by the vertices e i (1 i n) n ) , e i′ (1 i n) n ) and e and e i′′ (1 i n) n) in K in K 1,n,n,n and the vertices v vertices v,, e i induce a clique of order n + 1(say K n+1 ) in M ( M (K 1,n,n,n ). i.e., i.e.,V V ((M ( M (K 1,n,n,n)) = v vi : 1 i n wi : 1 i n ui : 1 i n ei : 1 i n ei′ : 1 i n ei′′ : 1 i n . Thus we have χ have χ r (M ( M (K 1,n,n,n )) n + 1.

≤ ≤



} {

Case 1.

≤ ≤



≤ ≤ } {

1

≤ ≤



≤ ≤ } { ≥



≤ ≤ } {

 

{ } { ≤ ≤ ≤ ≤ } { ≤ ≤ }

≤ r ≤ n. n .

Consider Consider the color class C class C 1 = c1 , c2 , c3 , , c(n+1) and assign the r-dynamic r -dynamic coloring to M ( M (K 1,n,n,n ) by Algorithm Algorithm 2.2.1. 2.2.1. Thus, Thus, an easy check check shows that the r adjacency condition is fulfilled. fulfilled. Hence, Hence, χ r (M ( M (K 1,n,n,n )) = n + n + 1, for 1 r n. n.

{

···

}

≤ ≤

Case 2.

r = n = n + 1.

−

103


{

···

}

Consider Consider the color class C class C 2 = c1 , c2 , c3 , , c(n+1) , c(n+2) and assign the r the r-dynamic -dynamic coloring to M ( M (K 1,n,n,n) by Algorithm Algorithm 2.2.2. Thus, Thus, an easy check check shows that the r adjacency condition is fulfilled. Hence, χ r (M ( M (K 1,n,n,n )) = n + 2, for r = n = n + 1 . Case 3.

−

r = ∆.

Consider the color class C 3 = c1, c2 , c3 , , cn , c(n+1) , c(n+2) , c(n+3) and assign the rdynamic coloring to M ( M (K 1,n,n,n ) by Algorithm Algorithm 2.2.3. Thus, Thus, an easy check check shows that the r adjacency condition is fulfilled. Hence, χ r (M (K 1,n,n,n )) = n + n + 3, for r ∆.

{

···

}

≥

Algorithm 2.2.1 Input: The Input: The number ”n ”n” of K K 1,n,n,n . Output: Assigning Output: Assigning r r-dynamic -dynamic coloring for the vertices in M ( M (K 1,n,n,n ). begin for i for i = 1 to n to n

{

{ }

V 1 = ei ; C (ei ) = i; i ;

}

{}


{

V 3 = vi ; C (vi ) = n + n + 1;

{ }

}


{

−1

V 4 = ei′ ; C (ei′ ) = i + 1;

{ }

}

C (e′n ) = 1; for i for i = 1 to n to n

{

−2

V 5 = wi ; C (wi ) = i + 2;

{ }

}

C (wn−1 ) = 1; C (wn ) = 2; for i for i = 1 to n to n

{

V 6 = ei′′ ; C (ei′′ ) = n + 1;

{ }

−

104


}


{

V 7 = ui ; C (ui ) = i; i ;

}

{ }

V = V 1 end

      V 2

V 3

V 4

V 5

V 6

V 7 ;


{

{ }

V 1 = ei ; C (ei ) = i; i ;

}

{}


{

{ }

V 3 = vi ; C (vi ) = n + n + 2;

}


{

V 4 = ei′ ; C (ei′ ) = n + 1;

{ }

}


{

−1

V 5 = wi ; C (wi ) = i + 1;

{ }

}

C (wn ) = 1; for i for i = 1 to n to n

{

−2

V 6 = ei′′ ; C (ei′′ ) = i + 2;

}

{ }

C (e′′n−1 ) = 1;


C (e′′n ) = 2; for i for i = 1 to n to n

{

V 7 = ui ; C (ui ) = n + 1;

}

{ }

V = V 1 end

      V 2

V 3

V 4

V 5

V 6

V 7 ;


{

{ }

V 1 = ei ; C (ei ) = i; i ;

}

{}


{

{ }

V 3 = vi ; C (vi ) = n + n + 2;

}


{

V 4 = ei′ ; C (ei′ ) = n + 3;

{ }

}


{

V 5 = wi ; C (wi ) = n + 1;

{ }

}


{

−1

V 6 = ei′′ ; C (ei′′ ) = i + 1;

}

{ }

C (e′′n ) = 1; for i for i = 1 to n to n

105

106


{

{ }

V 7 = ui ; C (ui ) = n + 2;

}

V = V 1 end

      V 2

V 3

V 4

V 5

V 6

V 7 ;

Theorem 2.3 For any triple star graph K 1,n,n,n , the r r -dynamic chromatic number,

χr (T ( T (K 1,n,n,n )) =

  

≤ r ≤ n r + 1, 1 , n + 1 ≤ r ≤ ∆ − 2 2n, r = ∆ − 1 2n + 1, 1, r ≥ ∆

n + 1, 1, 1

Proof By definition definition of total graph, each edge vvi , vi wi and wi ui be subdivided by the vertices e i (1 i n) n ) , e i′ (1 i n) n ) and e and e i′′ (1 i n) n) in K in K 1,n,n,n and the vertices v vertices v,, e i induce a clique of order n + 1(say K n+1 ) in T ( T (K 1,n,n,n ). i.e. i.e.,,V ( V (T ( T (K 1,n,n,n)) = v vi : 1 i n wi : 1 i n ui : 1 i n ei : 1 i n ei′ : 1 i n ei′′ : 1 i n . Thus, we have χ have χ r (T ( T (K 1,n,n,n)) n + 1.

≤ ≤



} {

Case 1.

≤ ≤



≤ ≤ } {

1

≤ ≤



 

{ } { ≤ ≤ ≤ ≤ } { ≤ ≤ }



≤ ≤ } { ≥

≤ ≤ } {

≤ r ≤ n. n .

Consider Consider the color class C class C 1 = c1 , c2 , c3 , , c(n+1) and assign the r-dynamic r -dynamic coloring to T ( T (K 1,n,n,n) by Algorithm 2.3.1. Thus, an easy check shows that the r adjacency condition is fulfilled. Hence, χ r (T ( T (K 1,n,n,n)) = n + n + 1, for 1 r n. n.

{

···

}

−

≤ ≤

Case 2.

n+1

≤ r ≤ ∆ − 2.

Consider the color class C class C 2 = c1 , c2 , c3 , , c(2n the r-dynamic -dynamic coloring to (2n−1) and assign the r T ( T (K 1,n,n,n) by Algorithm 2.3.2. Thus, an easy check shows that the r adjacency condition is fulfilled. Hence, χ r (T ( T (K 1,n,n,n)) = r + r + 1, for n + 1 r ∆ 2.

{

···

}

−

≤ ≤ −

Case 3.

r = ∆

− 1.

Consider the color class C 3 = c1 , c2 , c3 , , c2n if r = ∆ 1 and assign the r-dynamic coloring to T ( T (K 1,n,n,n ) by Algorithm 2.3.3. Thus, an easy check shows that the r adjacency condition is fulfilled. Hence, χ r (T ( T (K 1,n,n,n )) = 2n 2 n for r = ∆ 1.

{

···

}

−

−

Case 4.

−

r = ∆.

Consider the color class C 4 = c1 , c2 , c3 , , c2n+1 if r = ∆ and assign the r-dynamic coloring to T ( T (K 1,n,n,n ) by Algorithm 2.3.4. Thus, an easy check shows that the r adjacency condition is fulfilled. Hence, χ r (T ( T (K 1,n,n,n )) = 2n 2 n + 1 for r for r ∆.

{

···

}

≥

Algorithm 2.3.1 Input: The Input: The number ”n ”n” of K K 1,n,n,n .

−


Output: Assigning Output: Assigning r r-dynamic -dynamic coloring for the vertices in T ( T (K 1,n,n,n ). begin for i for i = 1 to n to n

{

{ }

V 1 = ei ; C (ei ) = i; i ;

}

{}


{

−3

V 3 = vi ; C (vi ) = i + i + 3;

{ }

}

C (vn−2 ) = 1; C (vn−1 ) = 2; C (vn ) = 3; for i for i = 1 to n to n

{

−2

V 4 = ei′ ; C (ei′ ) = i + 2;

{ }

}

C (e′n−1 ) = 1; C (e′n ) = 2; for i for i = 1 to n to n

{

−1

{ }

V 5 = wi ; C (wi ) = i + 1;

}


{

V 6 = ei′′ ; C (ei′′ ) = n + 1;

{ }

}


{

V 7 = ui ; C (ui ) = i; i ;

}

{ }

107

108


V = V 1 end

      V 2

V 3

V 4

V 5

V 6

V 7 ;

Algorithm 2.3.2 Input: The Input: The number ”n ”n” of K K 1,n,n,n . Output: Assigning Output: Assigning r r-dynamic -dynamic coloring for the vertices in T ( T (K 1,n,n,n ). begin for i for i = 1 to n to n

{

V 1 = ei ; C (ei ) = i; i ;

}

{ }


{

{}

−2

{ }

V 3 = vi ; C (vi ) = r + r + 1;

}

C (vn−1 ) = n + 2; C (vn ) = n = n + 3; for i for i = 1 to n to n 3

{

−

V 4 = ei′ ; C (ei′ ) = n + i + 2;

{ }

}

C (e′n−2 ) = n + n + 2; ′ C (en−1 ) = n + n + 3; ′ C (en ) = n + n + 4; for i for i = 1 to n to n 1

{

−

{ }

V 5 = wi ; C (wi ) = i + 1;

}


{

V 6 = ei′′ ; C (ei′′ ) = n + 1;

}

{ }



{

{ }

V 7 = ui ; C (ui ) = i; i ;

}

V = V 1 end

      V 2

V 3

V 4

V 5

V 6

V 7 ;

Algorithm 2.3.3 Input: The Input: The number ”n ”n” of K K 1,n,n,n . Output: Assigning Output: Assigning r r-dynamic -dynamic coloring for the vertices in T ( T (K 1,n,n,n ). begin for i for i = 1 to n to n

{

V 1 = ei ; C (ei ) = i; i ;

}

{ } {}


{

−1

V 3 = vi ; C (vi ) = n + n + i + 1;

{ }

}

C (vn ) = n = n + 2; for i for i = 1 to n to n 2

{

−

V 4 = ei′ ; C (ei′ ) = n + i + 2;

{ }

}

C (e′n−1 ) = n + n + 2; ′ C (en ) = n + n + 3; for i for i = 1 to n to n 1

{

−

{ }

V 5 = wi ; C (wi ) = i + 1;

}


{

V 6 = ei′′ ; C (ei′′ ) = n + 1;

}

{ }

109

110



{

{ }

V 7 = ui ; C (ui ) = i; i ;

}

V = V 1 end

      V 2

V 3

V 4

V 5

V 6

V 7 ;

Algorithm 2.3.4 Input: The Input: The number “n “n” of K K 1,n,n,n . Output: Assigning r Assigning r-dynamic -dynamic coloring for the vertices in T ( T (K 1,n,n,n ). begin for i for i = 1 to n to n

{

V 1 = ei ; C (ei ) = i; i ;

}

{ }


{

{}

{ }

V 3 = vi ; C (vi ) = n + n + i + 1;

}


{

−1

V 4 = ei′ ; C (ei′ ) = n + i + 2;

{ }

}

C (e′n ) = n + n + 2; for i for i = 1 to n to n 1

{

−

{ }

V 5 = wi ; C (wi ) = i + 1;

}


{

V 6 = ei′′ ; C (ei′′ ) = n + 1;

}

{ }


111


{

{ }

V 7 = ui ; C (ui ) = i; i ;

}

     

V = V 1 end

V 2

V 3

V 4

V 5

V 6

V 7 ;

Theorem 2.4 For any triple star graph K 1,n,n,n , the r r -dynamic chromatic number,

χr (L(K 1,n,n,n )) =

 

≤ r ≤ n − 1 r ≥ ∆

n, 1

n + 1, 1,

Proof First we apply apply the definition definition of line graph on K 1,n,n,n . By the definit definition ion of line line graph, each edge of K 1,n,n,n taken to be as vertex in L(K 1,n,n,n).The vertices e1 , e2 , , en induce a clique of order n in L(K 1,n,n,n). i.e., i.e., V ( V (L(K 1,n,n,n )) = E (K 1,n,n,n ) = ei : 1 i n ei′ : 1 i n ei′′ : 1 i n . Thus, we have χ r (L(K 1,n,n,n )) n. n .

} {

Case 1.

≤ ≤ } { 1

≤ ≤ }

··· ≤ ≤

{

≥

≤ r ≤ ∆ − 1.

Now consider consider the vertex set V ( V (L(K 1,n,n,n)) and color class C 1 = c1 , c2 , , cn , assign r assign r dynamic coloring to L(K 1,n,n,n) by Algorithm 2.4.1. 2.4.1. Thus, Thus, an easy check check shows that the r adjacency condition is fulfilled. Hence, χ r (L(K 1,n,n,n )) = n, n , for 1 r ∆ 1.

{

··· }

−

≤ ≤ −

Case 2.

r

≥ ∆. {

···

}

Now consider the vertex set V set V ((L(K 1,n,n)) and color class C class C 2 = c1 , c2 , , cn , cn+1 , assign r dynamic coloring to L( L (K 1,n,n,n ) by Algorithm 2.4.2. Thus, an easy check shows that the r adjacency condition is fulfilled. Hence, χ r (L(K 1,n,n,n )) = n + n + 1 for r ∆.

≥

Algorithm 2.4.1 Input: The Input: The number “n “n” of K K 1,n,n,n . Output: Assigning Output: Assigning r r-dynamic -dynamic coloring for the vertices in L( L (K 1,n,n,n ). begin for i for i = 1 to n to n

{

{ }

V 1 = ei ; C (ei ) = i; i ;

}


{

V 2 = ei′ ; C (ei′ ) = i + 1;

}

{ }

C (e′n ) = 1;

−1

−

112



{

−2

V 3 = ei′′ ; C (ei′′ ) = i + 2;

{ }

}

C (e′′n−1 ) = 1; C (e′′n ) = 2; V = V 1 V 2 V 3 ; end

 

Algorithm 2.4.2 Input: The Input: The number “n “n” of K K 1,n,n,n . Output: Assigning Output: Assigning r r-dynamic -dynamic coloring for the vertices in L( L (K 1,n,n,n ). begin for i for i = 1 to n to n

{

{ }

V 1 = ei ; C (ei ) = i; i ;

}


{

V 2 = ei′ ; C (ei′ ) = n + 1;

{ }

}


{

−1

V 3 = ei′′ ; C (ei′′ ) = i + 1;

{ }

}

C (e′′n ) = 1; V = V 1 V 2 end

 

V 3 ;

References [1] A. Ahadi, S. Akbari, A. Dehghana, M. Ghanbari, On the difference between chromatic number and dynamic chromatic number of graphs, Discrete Math. 312 (2012), 2579–2583. [2] S. Akbari, M. Ghanbari, Ghanbari, S. Jahanbak Jahanbakam, am, On the dynamic chromatic chromatic number number of graphs, Contemp. Math. (Amer. Math. Soc.), 531 (2010), 11–18. [3] S. Akbari, M. Ghanbari, Ghanbari, S. Jahanbekam, Jahanbekam, On the list dynamic coloring coloring of graphs, graphs, Discrete Appl. Math. 157 (2009), 3005–3007

113


Discrete ete Appl. Appl. Math. Math. 160 [4] M. Alishahi, Alishahi, Dynamic chromatic chromatic number of regular regular graphs, Discr (2012), 2098–2103. [5] J. A. Bondy, Bondy, U. S. R. Murty, Murty, Graph Theory , Springer, Springer, 2008. [6] Danuta Danuta Michalak, Michalak, On middle and total graphs with coarseness coarseness number number equal 1, Springer Springer Proceedings edings , Berlin Heidelberg, New York, Tokyo, (1981), Verlag, Graph Theory, Lagow Proce 139–150. [7] A. Dehghan, A. Ahadi, Upper bounds for the 2-hued chromatic chromatic number number of graphs in terms Discrete Appl. Math. 160(15) (2012), 2142–2146. of the independence number, Discrete [8] H. J. Lai, B. Montgomery, H. Poon, Upper bounds of dynamic chromatic number, Ars Combin. 68 (2003), 193–201. [9] X. Li, W. Zhou, The 2nd-orde 2nd-orderr conditiona conditionall 3-coloring 3-coloring of claw-fre claw-freee graphs, graphs, Theoret. Comput. Sci. 396 (2008), 151–157. [10] X. Li, X. Yao, W. Zhou, H. Broersma, Broersma, Complexity Complexity of condition conditional al colorability colorability of graphs, Appl. Math. Lett. 22 (2009), 320–324. [11] N. Mohanapriya, J. Vernold Vivin and M. Venkatachalam, Venkatachalam, δ δ -- dynamic chromatic number of helm graph families, Cogent Mathematics , 3(2016), No. 1178411. [12] N. Mohanapriy Mohanapriya, a, J. Vernold ernold Vivin and M. Venkatac enkatachalam halam,, On dynamic dynamic coloring coloring of Fan graphs, Int J of Pure Appl Math , 106(2016), 106(2016), 169-174. [13] N. Mohanapriya, Mohanapriya, Ph.D thesis, A Study on Dynamic Coloring of Graphs , Bharathiar University, (2017), Coimbatore, India. [14] B. Montgomery, Montgomery, Dynamic Coloring of Graphs , ProQuest LLC, Ann Arbor, MI, (2001), Ph.D Thesis, West Virginia University. Discrete Appl. Math., 201(2016), [15] A. Taherkhani, Taherkhani, r r-Dynamic -Dynamic chromatic number of graphs, Discrete 222–227. Harmonious Coloring of Total Graphs, Graphs, n Leaf, Central [16] J. Vernold Vernold Vivin, Ph.D Thesis, Harmonious Graphs and Circumdetic Graphs , Bharathiar University, (2007), Coimbatore, India. [17] A. T. White, White, Graphs, Groups and Surfaces , American Elsevier, New York, 1973.

−


(1,N)-Arithmetic Labelling of Ladder and Subdivision of Ladder V.Ramachandran (Department of Mathematics, Mannar Thirumalai Naicker College, Madurai, Tamil Nadu, India) E-mail: [email protected]

Abstract: Abstract: A ( p, q )-graph G )-graph G is said to be (1, (1, N )-arithmetic )-arithmetic labelling if there is a function φ function φ from the vertex set V set V ((G) to { to {0 0, 1, N, (N + + 1), 1), 2N, (2N (2N + + 1), 1), · · · , N (q − − 1), 1), N (q − − 1) + 1} 1} so that the values obtained as the sums of the labelling assigned to their end vertices, can be arranged in the arithmetic progression { progression {1 1, N + + 1, 2N + + 1, · · · , N (q − q − 1)+1} 1)+1}. In this paper we prove that ladder and subdivision of ladder are (1, (1, N )-arithmetic )-arithmetic labelling for every positive integer N integer N > 1. modulo N graceful, graceful, Smarandache k Smarandache k modKey Words: Words: Ladder, subdivision of ladder, one modulo N ulo N ulo N graceful. graceful.


V.Ramachandran and C. Sekar [8, 9] introduced one modulo N graceful N graceful where N where N is is any positive integer. integer. In the case N N = 2, the labelling is odd graceful and in the case N = N = 1 the labelling is grace gracefu ful. l. A graph graph G with q q edges is said to be one modulo N graceful graceful (where (where N is a positive integer) if there is a function φ from φ from the vertex set of G of G to to 0, 1, N, (N + + 1), 1), 2N, (2N (2N + + 1), 1), , N ( N (q 1), 1), N ( N (q 1) + 1 in such a way that (i ( i) φ is 1 1 (ii) ii) φ induces a bijection φ∗ from the edge set of G of G to 1, N + + 1, 2N + + 1, 1 , , N ( N (q 1) + 1 where φ where φ ∗ (uv) uv ) = φ(u) φ(v) . Generally, a graph G with q q edges is called to be Smarandache k modulo N graceful if one replacing N by kN kN in the definition of one modulo N N graceful graceful graph. Clearly Clearly, a graph G is Smarandache k Smarandache k modulo N modulo N graceful graceful if and only if it is one modulo kN k N graceful graceful by definition.

···

−

−

{

}

···

−

− }

{

|

−

|

B. D. Acharya and S. M. Hegde [2] introduced ( k, d)- arithmetic arithmetic graphs. graphs. A ( p, ( p, q ))- graph G is said to be (k, ( k, d)- arithmetic if its vertices can be assigned distinct nonnegative integers so that the values of the edges, obtained as the sums of the numbers assigned to their end vertices, can be arranged in the arithmetic progression k, k + d, k + 2d , . . . , k + (q 1)d 1)d. Joseph A. Gallian Gallian [4] surveyed numerous graph labelling methods.

−

V.Ramachandran and C. Sekar [10] introduced (1 , N )-Ari N )-Arithme thmetic tic labelling. labelling. We proved proved that stars, paths, complete bipartite graph K m,n m,n, highly irregular graph H i (m, m) and cycle C 4k are (1, (1, N )-Arithmetic N )-Arithmetic labelling,C labelling, C 4k+2 is not (1, (1 , N )-Ari N )-Arithme thmetic tic labelling. labelling. We also proved that no graph G graph G containing containing an odd cycle is (1, (1 , N )-arithmetic N )-arithmetic labelling for every positive integer 1 Received

August 24, 2017, Accepted May 28, 2018.

115

(1,N)-Arithmetic Labelling of Ladder and Subdivision of Ladder

N . N . A ( p, q )-graph G )-graph G is said to be (1 , N )-Arithmetic N )-Arithmetic labelling if there is a function φ : φ : V V ((G) 0, 1, N, (N + + 1), 1) , 2N, (2N (2N + + 1), 1) , , N ( N (q 1), 1), N ( N (q 1) + 1 .

{

···

−

−

}

→

In this situation the induced mapping φ∗ to the edges is given by φ∗ (uv)= uv)=φ φ(u) + φ( φ(v). If the values of φ(u) + φ( φ(v) are 1, 1, N + 1, 1 , 2N + 1, 1 , . . . , N ( q 1) + 1 all distinct, then we call the labelling of vertices as (1, (1 , N )N )- Arithmetic Arithmetic labelling. labelling. In case if the induced induced mapping φ∗ is defined as φ∗ (uv)= uv)= φ(u) φ(v) and if the resulting edge labels are are distinct and equal to 1, N + + 1, 2N + + 1, , N ( N (q 1) + 1 . We call it as one modulo modulo N N graceful. graceful. In this paper we prove prove that Ladder Ladder and Subdivision Subdivision of Ladder Ladder are (1, (1, N )-Arithmetic N )-Arithmetic labelling for every positive integer N > 1.

− −

| − | ··· − }

{

§2. Main Results

Definition 2.1 A graph G with q edges is said to be one modulo N graceful (where N is a positive integer) if there is a function φ from the vertex set of G to 0, 1, N, (N + + 1), 1), 2N, (2N (2N + + 1), 1), , N ( N (q 1), 1), N ( N (q 1 ) + 1 in such a way that (i) φ is 1 1 and (ii) ii) φ induces a bijection ∗ from the edge set of G φ from G to 1, N + + 1, 2N + + 1, , N ( N (q 1)+1 where φ φ ∗ (uv) uv ) = φ(u) φ(v) .

···

−

−

{

}

···

−

−

{

}

|

−

|

Definition 2.2 A ( p, ( p, q )-graph G is said to be (1, (1 , N ) N )-Arithmetic labelling if there is a function φ from the vertex set V ( V (G) to 0, 1, N, (N + N + 1), 1), 2N, (2N (2N + + 1), 1), , N ( N (q 1), 1), N ( N (q 1)+1 so that the values obtained as the sums of the labelling assigned to their end vertices, can be arranged in the arithmetic progression 1, N + + 1, 1 , 2N + + 1, 1 , , N ( N (q 1) + 1 .

{

···

{ {

···

−

−

−

}

}

graph G is said to be (k, d)- arithmetic if its vertices can be asDefinition 2.3 A ( p, q )- graph signe signed d distin distinct ct nonne nonnegat gativ ivee inte integer gerss so that that the the values values of the edges, dges, obtai obtaine ned d as the sums sums of the numbers numbers assigne assigned d to their their end vertice vertices, s, can can be arrange arranged d in the arithmet arithmetic ic pro progressio gression n k, k + d, k + 2d, 2 d, , k + (q ( q 1)d 1)d.

···

−

edges. A graph graph H is said to be a Definition 2.4([7]) Let G be a graph with p vertices and q edges. subdivision subdivision of G if H is obtained from G by subdividing every edge of G exactly once. H is denoted by S ( S (G). Definition 2.5 The ladder graph Ln is defined by Ln = P n K 2 where P n is a path with denotes the cartesian product. Ln has 2n vertices and 3n 2 edges.

−

×

× ×

Theorem 2.6 For every positive integer n, ladder Ln is (1, (1 , N ) N )-Arithmetic labelling, for every positive integer N > 1. 1 .

Proof Let u1 , u2 , , un and v1 , v2 , , vn be the vertices of Ln , respective respectively ly,, and let uivi+1 , i = 1, 2, , n 1. v i ui+1 , i = 1, 2, , n 1 and u and u i vi , i = 1, 2, , n be the edges of L L n . The ladder graph Ln is defined by L by Ln = P = P n K 2 where P where P n is a path with denotes the cartesian product. product. Then the the ladder ladder Ln has 2n 2nvertices and 3n 3 n 2 edges as shown in figures following. Define φ Define φ((ui ) = N ( N (i 1) for i for i = = 1, 2, 3, , n, φ( φ (vi ) = 2N ( N (i 1) + 1 for i = 1, 2, 3, , n.

··· ··· − −

··· ··· − ×

···

−

··· ×

−

···

116

V.Ramachandran

u1

v1

v2

u2

u3

v3

v4

u4

vn−1

un−1 vn

un

Figure 1 Ladder L Ladder L n where n where n is odd

u1

v1

v2

u2

u3

v3

v4

u4

un−1

vn−1 un

vn

Figure 2 Ladder L Ladder L n where n where n is even From the definition of φ of φ it is clear that



{φ(ui ), i = 1, 2, · · · , n} {φ(vi ), i = 1, 2, · · · , n} = {0, N, 2N , . . . , N (n − 1)} ∪ {1, 2N + + 1, 1 , 4N + + 1, 1 , · · · , 2N ( N (n − 1) + 1 } It is clear that the vertices have distinct labels. Therefore φ is φ is 1 labels as follows:

− 1. We compute the edge

for i = 1, 2, , n, φ∗ (vi ui ) = φ(vi ) + φ + φ((ui ) = 3N ( N (i 1) + 1; for i = 1, 2, , n 1, φ∗ (vi+1 ui ) = φ( φ(vi+1 ) + φ(ui ) = N = N (3 (3ii 1)+1, φ 1)+1, φ∗ (vi ui+1 ) = φ( φ(vi ) + φ(ui+1 ) = N = N (3 (3ii 2)+1.

···

|

|

− |

··· − | | − | − − 1) + 1}, This shows that the edges have the distinct labels {1, N + + 1, 2N + + 1, · · · , N ( N (q − where q where q = = 3n − 2. Hence L Hence L n is (1, (1, N )-Arithmetic N )-Arithmetic labelling for every positive integer N > 1. Example 2.7. A (1, (1, 5)-Arithmetic labelling of L of L 6 is shown in Figure 3.

117


0

1

11

5

10

21

31

15

20

41

51

25 Figure 3

Example 2.8 A (1, (1, 2)-Arithmetic labelling of L of L 7 is shown in Figure 4. 0

1

5

2

4

9

13

6

8

17

21

10

12

25 Figure 4

-Arithmetic labelling for every positive positive integer Theorem 2 Theorem 2..9 A subdivision of ladder L n is (1, (1 , N ) N )-Arithmetic N > 1. 1 .

Proof Let G Let G = L = L n . The ladder graph L n is defined by Ln = P n K 2 where P where P n is a path with denotes the cartesian product. Ln has 2n 2n vertices and 3n 3 n 2 edges. A graph H graph H is is said to be a subdivision of G of G if H if H is is obtained from G by subdividing every edge of G G exactly once. H is is denoted by S ( S (G). Then the subdivision subdivision of ladder ladder L n has 5n 5n 2 vertices and 6n 6 n 4 edges as shown in Figure 5. Let H Let H = = S (Ln ).

×

−

−

×

−

118

V.Ramachandran

u1 u12 u2 u23 11

u3

u34 u4

v2 v23 v3

w3 w4

u56

w5

u6

v12

w2

u45 u5

v1

w1

v34 v4 v45 v5 v56 v6

w6

un−1

vn−1

wn−1 un−1,n vn−1,n

un

vn

wn

Figure 5 Subdivision of ladder L n Define the following functions: η : N : N

→ N by → η (i) =

and γ and γ : : N N

→ N by → γ (i) =

→ {0, 1, 2, · · · , q } by

Define φ Define φ : : V V

   

− 1) 2N ( N (i − 1)

if i is even

− 1) N (2 N (2ii − 1)

if i is even

N (2 N (2ii

2N ( N (i

if i is odd

if i is odd

φ(ui ) = η = η((i), i = 1, 2,

· · · , n φ(vi ) = γ = γ (i), i = 1, 2, · · · , n.

Define φ(ui,i+1 i,i+1 ) =

 

1 + (i (i

− 1)4N 1)4N

− 1)N 1)N + 1 For i or i = 1, 2, · · · , n − 2, define (4i (4i

if i i is odd if i is even.

φ(vi,i+1 φ(ui+1,i i,i+1 ) = φ( +1,i+2 +2 )

− 4N,

φ(vn−1,n ) = φ( φ(un−2,n− 4 N, ,n−1 ) + 4N,

119


φ(wi ) =

 

1 + (4i (4 i 4N n

− 3)N 3)N

if i = i = 1, 2,

− 4N + 1

if i = i = n n .

· · · , n − 1

It is clear that the vertices have distinct labels. Therefore φ is φ is 1 labels as follows:

− 1. We compute the edge

φ∗ (wn un ) = φ(wn ) + φ(un ) = 6N n

| | − 6N + + 1, 1, φ∗ (wn vn ) = | φ( φ(wn ) + φ(vn ) | = 6N n − 5N + + 1, 1, φ∗ (vn−1,n vn−1 ) = φ( φ(vn−1,n ) + φ(vn−1 ) =

|

|

φ∗ (vn−1,n vn ) = φ( φ(vn−1,n ) + φ(vn ) =

|

For i or i = 1, 2,

|

· · · , n − 1,

φ∗ (wi ui ) = φ(wi ) + φ(ui ) =

|

|

φ∗ (wi vi ) = φ( φ(wi ) + φ(vi ) =

|

For i or i = 1, 2,

|

· · · , n − 1,

   

if i i is even

 

|

· · · , n − 2,

φ∗ (vi,i+1 φ(vi,i+1 i,i+1 vi ) = φ( i,i+1 ) + φ(vi ) =

|

 

if i i is odd.

− 2) + 1 N (6 N (6ii − 6) + 1

|

− 6) + 1 N (6 N (6ii − 2) + 1

 

if i i is even if i i is odd.

− 3) + 1 N (6 N (6ii − 1) + 1 N (6 N (6ii

N (6 N (6ii

,

if i i is odd.

N (6 N (6ii

 

if n is even.

if n n is even.

− 5) + 1 N (6 N (6ii − 4) + 1 N (6 N (6ii

if n n is odd

if n n is odd

if i i is even

|

|

− 9N + 1 6N n − 7N + 1 6N n

− 4) + 1 N (6 N (6ii − 5) + 1

φ∗ (ui,i+1 φ(ui,i+1 i,i+1 ui+1 ) = φ( i,i+1 ) + φ(ui+1 ) = For i or i = 1, 2,

− 12 12N N + 1 6N n − 8N + 1 6N n

N (6 N (6ii

φ∗ (ui,i+1 φ(ui,i+1 i,i+1 ui ) = φ( i,i+1 ) + φ(ui ) =

|

 

 

if i i is odd if i i is even.

if i i is even if i i is odd.

− 3) + 1 if i i is even N (6 N (6ii − 1) + 1 if i i is odd. This shows that the edges have distinct labels {1, N + + 1, 2N + + 1, 1 , · · · , N ( N (q − − 1) + 1} with q = = 6n − 4. Hence S Hence S ((Ln) is (1, (1, N )-Arithmetic N )-Arithmetic labelling for every positive integer N > 1. φ∗ (vi,i+1 φ(vi,i+1 i,i+1 vi+1 ) = φ( i,i+1 ) + φ(vi+1 ) =

|

|

N (6 N (6ii

Example 2.10 A (1, 3)-Arithmetic labelling of S of S ((L5 ) is shown in Figure 6.

120

V.Ramachandran

4

0

3

1

10 16

9

6

22

13 28

12

15

25

34 40

21

18 37

46

24

27

49 Figure 6

Example 2.11 A (1, 10)-Arithmetic labelling of S of S ((L6 ) is shown in Figure 7. 0

11

1 30

51

91 40

41 50

131

70 151

31 20

71

81

10

111 60

171

121

80

90

161

191

110

201 Figure 7

100


121

References [1] B. D. Acharya, Acharya, On d-sequent d-sequential ial graphs, J. Math. Phys. Sci. , 17 (1983), 21-35. [2] B. D. Acharya Acharya and S. M. Hegde, Hegde, Arithmetic Arithmetic graphs, Journal of Graph Theory , 14 (1990) 275-299. [3] B. D. Acharya Acharya and S. M. Hegde, Hegde, On certain certain vertex valuat valuations ions of a graph I, Indian J. Pure Appl. Math., 22 (1991) 553-560. [4] Joseph A. Gallian, Gallian, A Dynamic Dynamic Survey of Graph Labeling, Labeling, The Electronic Journal of Combinatorics , #DS6 (2016). [5] S.W.Golomb, S.W.Golomb, How to Number a Graph in Graph theory and Computing , R.C. Read, ed., Academic Press, New york (1972)23-27. [6] R. B. Gnanajothi, Topics in Graph theory , Ph.D. thesis, Madurai Madurai Kama Kamaraj raj Universit University y, 1991. [7] KM. Kathiresan, Subdivisions Subdivisions of ladders are graceful, graceful, Indian J. Pure Appl. Math. , (1992), 21–23. [8] V. Ramachandran, C. Sekar, Sekar, One modulo N gracefullness of arbitrary supersubdivisions of International J. Math. Combin., Vol.2 (2014), 36-46. graphs, International [9] V. Ramachandran, C. Sekar, One modulo N gracefulness of supersubdivision of ladder, Journal of Discrete Mathematical Sciences and Cryptography , Vol.18 (3) (2015), 265-274. [10] V. Ramachandran, C. Sekar, Sekar, (1, (1, N )-arithmetic N )-arithmetic graphs, International Journal of Computers and Applications Applications , Vol.38 (1) (2016) 55-59. [11] C. Sekar, Sekar, Studies in Graph Theory , Ph.D. thesis, Madurai Kamaraj University, 2002.


3-Difference Cordial Labeling of Corona Related Graphs R.Ponraj Department of Mathematics, Sri Paramakalyani College, Alwarkurichi-627412, India

M.Maria Adaickalam Adaickalam Department of Mathematics Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli-627012,Tamilnadu India E-mail: ponra [email protected], [email protected] [email protected]

Abstract: Abstract: Let G be a ( p, q ) graph. graph. Let f Let f : V ( V (G) → {1, 2, · · · , k} be a map where k is an integer 2 ≤ 2 ≤ k ≤ p. p . For each edge uv, uv, assign the label |f (u) − f (v)|. f f is called k-difference cordial labeling of G if |vf (i) − vf ( j) j )| ≤ 1 and |ef (0) − ef (1)| (1)| ≤ 1 where v where v f (x) denotes the number of vertices labelled with x with x,, e f (1) and e and e f (0) respectively denote the number of edges labelle labelled d with with 1 and not labelle labelled d with with 1. A graph with with a k-differenc -differencee cordial cordial labeling labeling is called a k a k-difference -difference cordial graph. In this paper we investigate 3-difference cordial labeling behavior of DT DT n ⊙ K 1 DT n ⊙ 2K 1 ,DT n ⊙ K 2 and some more graphs.

Key Words: Words: Difference cordial labeling, Smarandachely k Smarandachely k-difference -difference cordial labeling, path, complete graph, triangular snake, corona.


All Graphs in this paper are finite ,undirect ,undirect and simple. Let G1 , G2 respectively be ( p ( p1 , q 1 ), ( p2 , q 2 ) graphs. The corona of G G 1 with G with G 2 , G1 G2 is the graph obtained by taking one copy of G1 and p and p 1 copies of G G 2 and joining the i th vertex of G G 1 with an edge to every vertex in the i th copy of G G2 . Ponraj et al. [3], has been introduced the concept of k of k-difference -difference cordial labeling of graphs and studied the 3-difference cordial labeling behavior of of some graphs. In [4,5,6,7] they investigate the 3-difference cordial labeling behavior of path, cycle, complete graph, complete (t) bipartite graph, star, bistar, comb, double comb, quadrilateral snake, C 4 , S (K 1,n ), S (Bn,n ) and carona of some graphs with double alternate triangular snake double alternate quadrilateral snake snake . In this paper paper we examine examine the 3-diffe 3-differen rence ce cordial cordial labeling labeling behavior behavior of DT n K 1 DT n 2K 1 ,DT n K 2 etc. Terms are not defined here follows from Harary [2].

⊙

⊙

⊙

⊙

§2. k -Difference Cordial Labeling

Definition 2.1 Let G be a ( p, q ) graph and let f : V ( V (G) 1 Received Received November November


map. → {1, 2, · · · , k} bebe a map.

For

123

3-Difference Cordial Labeling of Corona Related Graphs

each edge uv , assign the label f ( f (u) f ( f (v) . f is called a k -difference cordial labeling of G if vf (i) vf ( j) j ) 1 and ef (0) ef (1) 1 where vf (x) denotes the number of vertices labelled with x x , e f (1) and ef (0) respectively denote the number of edges labelled with 1 and not labelled with 1 1 . A graph with a k -difference cordial labeling is called a k-difference cordial graph. On the other hand, if vf (i) vf ( j) j ) 1 or ef (0) ef (1) 1 , such a labeling is called a Smarandachely k -difference cordial labeling of G.

|

−

|≤

| | −

| |

− | |≤

| |

−

|≥

| |

−

|≥

A double triangular snake DT snake DT n consists of two triangular snakes that have a common path. That is a double triangular snake is obtained from a path u 1 u2 joining u i and u i+1 to un by joining u two new vertices v i (1 i n 1) and w and w i (1 i n 1). First we investigate the 3-difference cordial labeling behavior of D of DT T n K 1 .

≤ ≤ −

···

≤ ≤ −

⊙

⊙ K 1 is 3-difference cordial. Proof Let V ( V (DT n ⊙ K 1 ) = V ( V (DT n ) {xi : 1 ≤ i ≤ n} {vi′ , wi′ : 1 ≤ i ≤ n − 1 } and E (DT n ⊙ K 1) = E (DT n ) {uixi : 1 ≤ i ≤ n} {vivi′ , wi wi′ : 1 ≤ i ≤ n − 1}.

Theorem 2.1 DT n





Case 1. n is even.



First we consider consider the path vertices vertices ui . Assi Assign gn the label label 1 to all the path path vert vertic ices es ui (1 i n). assign the label 2 to the path vertice verticess v 1 , v3 , v5 , and assign the label 1 n ). Then assign to the path vertices v2 , v4 , v6 , . Now we consider consider the vertices vertices wi . Assign Assign the the label 2 to all ′ ′ the vertices wi (1 i n 1). Next we move move to the vertices vertices vi and wi . Assign Assign the the label 2 ′ to the vertices v 2i+1 for all the values of i of i = 0, 1, 2, 3, and assign the label 1 to the vertices v2i for i = 1, 2, 3, . Next Next we assign the the label label 1 to the vertex vertex w1′ and assign the label 3 to the vertices w 2′ , w3′ , w4′ , Finally assign the label 3 to all the vertices of x of x i (1 i n). n ). The 6n−3 6n−2 vertex condition and the edge conditions are vf (1) = vf (2) = 3 and vf (3) = 3 and ef (0) = 4n 4n 4 and e and e f (1) = 4n 4n 3.

≤ ≤

≤ ≤ − ···

···

···

···

···

−

≤ ≤

−

Case 2. n is odd.

≤ ≤

≤ ≤ −

≤ ≤ −

Assign the label to the path vertices u i (1 i n), n ), v v i (1 i n 1), w 1), w i (1 i n 1), (1 i n 1), xi (1 i n) as in case 1. Then Then assign the the label 3 to all the vertice verticess (1 i n 1). Since Since ef (0) = 4n 4 n 3, ef (1) = 4n 4n 4 and vf (1) = vf (3) = 2n 2n 1 and vf (2) = 2n 2 n 2, D 2, DT T n K 1 is 3-difference cordial.

vi′ wi′

≤ ≤ − ≤ ≤ − −

≤ ≤

⊙

Next investigation about DT D T n

−

−

−

⊙ 2K 1.

⊙ 2K 1 is 3-difference cordial. Proof Let V Let V ((DT n ⊙ 2K 1) = V ( V (DT n ) {xi, yi : 1 ≤ i ≤ n } {vi′ , vi′′ , wi′ , wi′′ : 1 ≤ i ≤ n − 1} and E (DT n ⊙ 2K 2 K 1 ) = E (DT n ) {ui xi , ui yi : 1 ≤ i ≤ n} {vi vi′ , vi vi′′ , wi wi′ , wi wi′′ : 1 ≤ i ≤ n − 1} .

Theorem 2.2 DT n





 

Case 1. n is even. Consider the path vertices ui . Assign Assign the the label 1 to the path path vertex vertex u1 . No Now w we assign assign the labels 1,1,2,2 to the vertices u 2 , u3 , u4 , u5 respectively. Then we assign the labels 1,1,2,2 to

124

R.Ponraj and M.Maria Adaickalam Adaickalam

the next four vertices u 6 , u7 , u8, u9 respectively. Proceeding like this we assign the label to the next four vertices and so on. If all the vertices are labeled then we stop the process. Otherwise there are some non labeled vertices vertices are exist. exist. If the number number of non labeled vertice verticess are less than than or equal to 3 then then assign assign the labels labels 1,1 1,1,2 ,2 to the non labeled labeled vertice vertices. s. If it is two then then assign assign the label label 1,1 to the non labeled labeled verti vertices ces.. If only one non labeled labeled vertex vertex is exist then assign assign the label 1 only only. Next Next we conside considerr the label vi . Assign Assign the label label 2 to the verte vertex x v1 . Then we assign the label 2 to the vertices v2 , v4 , v6 , and assign the label 3 to the vertices v3 , v5 , v7 , . Next we move to the vertices x i and y i . Assign the label 2 to the vertices x 1 and a nd x2 and we assign the label 3 to the vertices vertices y 1 and y 2 . Now we assign the label 1 to the vertices x4i+1 and x4i for all the values of i = 1, 2, 3, . Then Then we assign assign the label label 1 to the vertice verticess x4i+3 for i = 0, 1, 2, 3, . Next Next we assign assign the label 2 to the vertice verticess x4i+2 for all the values of i = 1, 2, 3, . Now we assign assign the label 3 to the verti vertices ces y4i+3 for i=0,1,2,3,.. i=0,1,2,3,.... For all the values of i i = 1, 2, 3, . assign the label 3 to the vertices y 4i+1 and y and y 4i+2 . Then we assign the label 2 to the vertices y4i for i = 1, 2, 3, . Now we consider consider the vertices vertices v v i′ and vi′′ . For all the values of i=1,2,3... assign the label 1 to the vertices v 4′ i+1 , v4′ i+2 . Assign Assign the label label 1 to the vertices v 4′ i for i for i = = 1, 2, 3, . Then we assign the label 2 to the vertices v 4′ i+3 for all the values of i i = 0, 1, 2, 3, . Consider the vertices v i′′ . Assign the label 3 to the path vertex v 4′′i+1 , v4′′i+2 and v4′′i+3 for all the values of i = 0, 1, 2, 3, . Next we assign assign the label 2 to the vertices vertices v4′′i for i for i = 1, 2, 3, . Now we assign the label 3 to the vertices w i (1 i n 1). Next we move to the vertices wi′ and w and w i′′ . Assign Assign the label label 1 to all the vertice verticess of w of w i′ (1 i n 1) and we assign the label 2 to all the vertices of w of w i′′ (1 i n 1). Since v Since v f (1) = v = v f (2) = v = v f (3) = 3n 3n 2 11n 11n−10 11n 11n−8 and e and e f (0) = and e and e f (1) = 2 , this labeling is 3-difference 3-difference cordial labeling. 2

···

···

···

···

···

···

···

···

···

···

···

≤ ≤ − ≤ ≤ −

≤ ≤ −

−

Case 2. n is odd. First we consider the path vertices ui . Assi Assign gn the label label 1,1,2 1,1,2,2 ,2 to the the first first four path path vertices u1 , u2 , u3 , u4 respective respectively ly.. Then we assign the labels 1,1,2,2 to the next four vertices vertices u5 , u6 , u7 , u8 respectively. Continuing like this assign the label to the next four vertices and so on. If all the vertices vertices are labeled then we stop the process. Otherwise Otherwise there there are some on labeled vertices are exist. If the number of non labeled vertices are less than or equal to 3 then assign the labels labels 1,1 1,1,2 ,2 to the non labeled labeled verti vertices ces.. If it is 2 assign assign the labels labels 1,1 to the non labeled labeled vertices. vertices. If only one non labeled labeled vertex exist then assign the label 1 to that vertex. vertex. Consider Consider the vertices vi . Assign Assign the the label 2 to the vertic vertices es v1 , v3 , v5 , and we assign the label 3 to the vertices v2 , v4 , v6 , . Next we move to the vertices wi . Assign Assign the label to the vertice verticess wi (1 i n ) as in case 1. Now we conside considerr the verti vertices ces xi and yi . Assign Assign the the label 2 to the vertices x vertices x 4i+1 for all the values of i of i = = 0, 1, 2, 3, . For all the values values of i=0,1,2,3, i=0,1,2,3,... ... assign the label 1 to the vertices x 4i+2 and x4i+3 . Then Then we assign assign the label label 1 to the vertice verticess x 4i for all the values of i = 1, 2, 3, . Next Next we assign assign the label label 3 to the vertic vertices es y 4i+1 and y4i+2 for all the values of i of i = = 0, 1, 2, 3, and we assign the label 3 to the vertices y 4i for i for i = = 1, 2, 3, . Then we assign the label 2 to the vertices y4i+3 for all values i = 0, 1, 2, 3, . Next we we move move to the vertices v i′ and v and v i′′ . For all the values of i of i = = 0, 1, 2, 3, assign the label 1 to the vertices ′ ′ v4i+1 and v 4i+3 . Now we assign the label 1 to the vertices v 4′ i for i = i = 1, 2, 3, . Next we assign ′ the label 2 to the vertices v 4i+2 for i for i = 01 01,, 2, 3, . Consider Consider the vertices vertices v v i′ . Assign Assign the the label

≤ ≤ −

···

···

···

··· ···

···

···

···

···

···

125


3 to the vertices v4′′i+1 and v4′′i+2 for all the values of i = 0, 1, 2, 3, and we assign the label 1 to the vertices v 4i for i for i = 1, 2, 3, . For the values values of i i = 0, 1, 2, 3, assign the label 2 to ′ the vertices v vertices v 4i+3 . Finally we consider the vertices w i and w and w i′′ . Assign the label to the vertices wi′ (1 i n 1) and wi′′ (1 i n 1) as in case 1. The vertex vertex and edge conditi condition on are 11n 11n−9 vf (1) = v = v f (2) = v = v f (3) = 3n 3 n 2 and e and e f (0) = e = e f (1) = 2 .

···

···

≤ ≤ −

−

···

≤ ≤ −

We now investigate the graph DT n

⊙ K 2.

⊙ K 2 is 3-difference cordial. Proof Let V Let V ((DT n ⊙ K 2 ) = V ( V (DT n ) {xi , yi : 1 ≤ i ≤ n } {vi′ , vi′′ , wi′ , wi′′ : 1 ≤ i ≤ n − 1} and E (DT n ⊙K 2 ) = E (DT n ) {ui xi , ui yi , xi yi : 1 ≤ i ≤ n} {vi vi′ , vi vi′′ , vi′ vi′′ , wi wi′ , wi wi′′ , wi′ wi′′ : 1 ≤ i ≤ n − 1}.

Theorem 2.3 DT n





 

Case 1. n is even. Consider the path vertices u i. Assign the label 1 to the path vertices u 1 , u2 , u3 , . Then we assign the labels 2 to the vertices v 1 , v2 , v3 , . Next we assign the labels 3 to the vertices w1 , w2 , w3 , w4 . Now we consider consider the vertices vertices vi′ and vi′′ . Assign Assign the label label 2 to the vertex vertex v1′ . Then we assign the label 1 to the vertices v2′ , v3′ , v4′ , v6′ , . No Now w we assign assign the label label 3 to the vertices v 1′′ , v2′′ , v3′′ , v4′′ , . Next we move to the vertices w i′ and w and w i′′ . Assign Assign the the label 1 to the ′ ′ ′ ′ vertex w1 . Then we assign the the label 1 to the vertices vertices w 2 , w4 , w6 , and assign the label 2 to ′ ′ ′ the vertices w3 , w5 , w7 , . Assign the label 2 to the vertices w1′′ , w2′′ , w3′′ , w4′′ , . Finally we move to the vertices x i and yi . Assign the the label 1 to the vertices vertices x 1 , x3 , x5 , and we assign the label 2 to the vertices x 2 , x4 , x6 , then we assign the label 3 to the vertices vertices y y 1 , y2 , y3 , . Clearly in this case the vertex and edge condition is given in vf (1) = v = v f (2) = v = v f (3) = 3n 3n 2 and e and e f (0) = 7n 7 n 5 and e and e f (1) = 7n 7 n 6.

···

···

···

···

···

···

···

−

··· ···

··· −

−

Case 2. n is odd. Assign the label to the vertices u i (1 i n), n), v v i (1 i n 1) and w and w i (1 i n 1) as in case 1. Consider Consider the vertices vertices v i′ and v and v i′′ . Assign the label 1 to the vertices v 1′ , v2′ , v3′ , v4′ , . Then assign the label to the vertices vi′′ (1 1) as in case case 1. No Now w we mo move ve to the i n 1) vertices w i′ and w and w i′′ . Assign the label 1 to the vertices w 1′ , w3′ , w5′ , and we assign the label 3 to the vertices w vertices w 2′ , w4′ , w6′ , . Next we assign the label to the vertices w i′′ (1 i n 1) as in case 1. Now we consider the vertices x i and y and y i . Assign the label 2 to the vertices x 1 , x3 , x5 , and we assign the label 1 to the vertices x 2 , x4 , x6 , . Then we assign the label to the vertices yi (1 i n) as in case 1. Sinc Sincee vf (1) = vf (2) = vf (3) = 3n 3 n 2 and ef (0) = 7n 7 n 6 and ef (1) = 7n 7n 5, this labeling is 3-difference cordial labeling.

≤ ≤

≤ ≤ −

≤ ≤ −

···

···

≤ ≤ −

···

≤ ≤ −

≤ ≤ −

−

···

···

−

A double quadrilateral snake DQ snake DQ n consists of two quadrilateral snakes that have a common path path.. Let Let V ( V (DQ n ) = ui : 1 i n vi , wi ,xi,yi : 1 i n 1 and E (DQn ) = ui ui+1 , vi wi , xi yi , wi ui+1 , y i ui+1 : 1 i n 1 .

{



≤ ≤ } { ≤ ≤ − } { ≤ ≤ − } Now we investigate the graphs DQ n ⊙ K 1 ,DQn ⊙ 2K 1 and D and DQ Qn ⊙ K 2 .

Theorem 2.4 DQ n

⊙ K 1 is 3-difference cordial.

126


Proof Let V ( V (DQn K 1 ) = V ( V (DQn ) ui′ : 1 i n vi′ , wi′ , xi′ , yi′ : 1 i n 1 ′ ′ ′ ′ ′ and E (DQn K 1 ) = E ( E (DQn ) ui ui : 1 i n ui vi , wi wi , xi xi , yi , yi : 1 i n 1 . Assign the label 1 to the path vertex u1 . Next Next we assi assign gn the label labelss 1,1, 1,1,22 to the verti vertice cess u2 , u3 , u4 respectively. Then we assign the labels 1,1,2 to the next three path vertices u 5 , u6 , u7 respective respectively ly.. Proceeding Proceeding like this we assign the label to the next three vertices vertices and so on. If all the vertices are labeled then we stop the process. Otherwise there are some non labeled vertices are exist. If the number of non labeled vertices vertices are less than or equal to 2 then assign the the labels 1,1 to the non labeled vertices. If only one non labeled vertex exist then assign the label 1 only. Now we consider the vertices v i and w and w i . Assign Assign the label 2 to the vertices vertices v 3i+1 and v and v 3i+2 for all the values of i=0,1,2,3... For all the vales of i of i = 1, 2, 3, assign the label 1 to the vertices v3i . Then we assign the label 3 to the vertices w i (1 i n). n ). Next we move to the vertices x i and y and y i . Assign the labels 2,3 to the vertices x 1 and y and y 1 respectively. Then we assign the label 2 to the vertices x 2 , x5 , x8 , . Now we assign the label 1 to the vertices x3 , x6 , x9 , and the vertices x vertices x4 , x7 , x10 , . Assign . Assign the label 3 to the vertices y1 , y2 , y3 , . We . We consider the vertices ′ ′ ′ ui. Assign Assign the labels 2,3 to the vertices vertices u u1 and u and u2 respectively. Now we assign the label 1 to the ′ vertices u 3′ , u6′ , u9′ , and we assign the label 3 to the vertices u 4′ , u7′ , u10 , . Then we assign ′ ′ ′ the label 2 to the vertices u 5 , u8 , u11 , . Next we move move to the vertices vertices v i′ and w and w i′ . Assign the the label 3 to the vertex w1′ . No Now w assign assign the label 1 to all the the vertices vertices of vi′ (1 i n 1) and we assign the label 2 to the vertices w2′ , w3′ , w4′ , . We consider the vertices xi′ and yi′ . Assign the label 2,1 to the vertices x1′ and y1′ respective respectively ly.. Also we assign the label 2 to the vertices x2′ , x5′ , x8′ , and the vertices x3′ , x6′ , x9′ , . Then we assign the label 1 to the vertices ′ x4′ , x7′ , x10 , . Next we assign the label 3 to the vertices y2′ , y3′ , y4′ ,... The vertex condition is ef (0) = 6n 6n 5 and e and e f (1) = 6n 6n 6. Also the edge condition is given in Table 1 following.

⊙

⊙

{ ≤ ≤ } { ≤ ≤ } {

{

≤ ≤ − } ≤ ≤ − }

··· ≤ ≤

···

···

···

···

···

···

···

≤ ≤ −

···

··· −

···

···

−

Nature of n n

vf (1) 10n 10n−9 3 10n 10n−10 3 10n 10n−8 3

≡ 0 (mod 3) n ≡ 1 (mod 3) n ≡ 2 (mod 3) n

vf (2) 10n 10n−6 3 10n 10n−7 3 10n 10n−8 3

vf (3) 10n 10n−9 3 10n 10n−7 3 10n 10n−8 3

Table 1 Theorem 2.5 DQ n

⊙ 2K 1 is 3-difference cordial.

Proof Let V Let V ((DQ n 2K 1 ) = V ( V (DQn ) ui′ , ui′′ : 1 i n vi′ , vi′′ , wi′ , wi′′ , xi′ , xi′′ , yi′ , yi′′ : 1 i n 1 and E (DQ n 2K 1 ) = E = E (DQn ) ui ui′ , ui ui′′ : 1 i n vi vi′ , vi vi′′ , wi wi′ , wi wi′′ , xi xi′ , xi xi′′ , y , y i yi′′ , yi yi′′ : 1 i n 1 . First we consider the path vertices u i . Assign the label 1 to the path vertices u 1 , u3 , u5 , and we assign the label 2 to the path vertices u 2 , u4 , u6 , . Clearly the last vertex u n received the label 2 or 1 according as n 0mod 2 or n or n 1 (mod 2). Next we move to the vertices v i and w and w i . Assign the label 1 to all the vertices of v of v i (1 i n) n ) and we assign the label 3 to the vertices w 1 , w2 , w3 ,... Then ,... Then we assign the label to the vertices xi (1 i n 1) is same as assign the label to the vertices vi (1 i n 1) and we assign the label to the vertices y i (1 i n 1) is same as assign the label to the vertices w i (1 i n 1). Next we move move to the vertices vertices ui′ and vi′′ . Assign Assign the label label 2 to the vertice verticess

≤ ≤ − }

≤ ≤ −

≤ ≤ −

⊙

⊙ ≤ ≤ −} ···

{ {

≤ ≤ −

 

≤ ≤ } { ≤ ≤ } { ≡

≡

≤ ≤ −

···

≤ ≤

127


v1′ , v2′ , v3′ , then we assign the label 3 to the vertex v1′′ . Assign the label 3 to the vertices v2′′i for all the values of i of i = = 1, 2, 3, and we assign the label 2 to the vertices v 2′′i+1 for i for i = = 1, 2, 3, . ′ ′′ ′ ′ Next we consider the vertices w i and w and w i . Assign the label 1 to the vertices w 1 , w2 , w3′ , and ′′ ′′ ′′ ′ we assign the label 3 to the vertices w1 , w2 , w3 , . Next we move to the vertices x i and xi′′ . Assign the label 1 to all the vertices of xi′ (1 i n 1) and we assign the label 2 to all the vertices of xi′′ (1 i n 1). No Now w we assign assign the label 2 to the verti vertices ces y1′ , y2′ , y3′ , and we assign the label 3 to the vertices y 1′′ , y2′′ , y3′′ , . Finally we move to the vertices u i′ and a nd ′′ ′ ′ ′ ui . Assign Assign the label 2 to the vertices vertices u1 , u3 , u5 , and we assign the label 1 to the vertices ′ ′ ′ u2 , u4 , u6 , . Next we assign the label 2 to the vertices u 1′′ , u3′′ , u5′′ , and we assign the label 15n−12 3 to the vertices u 2′′ , u4′′ , u6′′, . The vertex condition is v f (1) = v = v f (2) = v = v f (3) = 15n . Also Also 3 the edge condition is given in Table 2.

···

···

··· ≤ ≤ −

≤ ≤ −

···

···

···

··· ···

···

···

Values of n n

ef (0)

ef (1)

17n 17n−16 2 17n 17n−15 2

n

≡ 0 (mod 2) n ≡ 1 (mod 2)

···

17n 17n−14 2 17n 17n−15 2

Table 2 Theorem 2.6 DQ n

⊙ K 2 is 3-difference cordial.

Proof Let V Let V (DQn K 2 ) = V ( V (DQn ) ui′ , ui′′ : 1 i n vi′ , vi′′ , wi′ , w i′′ , xi′ , xi′′ , yi′ , yi′′ : 1 i n 1 and E and E (DQn K 2 ) = E (DQn ) uiui′ , ui ui′′ , ui′ ui′′ : 1 i n vivi′ , vi vi′′ , vi′ vi′′ , wi wi′ , wi wi′′ , wi′ wi′′ , xi xi′ , x i xi′′ , xi′ xi′′ , yi yi′′ , yi yi′′ , yi′ yi′′ : 1 i n 1 . First we consider the path vertices ui . Assign Assign the label label 1 to the verte vertex x u1 . Then Then we assign the the label 1 to the verti vertices ces u 2 , u4 , u6 , and we assign the label 2 to the path vertices u1 , u3 , u5 , . Note that in this case the last vertex u n received the label 1or 2 according as n 0 (mod 2) or n 1 (mod 2). Next we move to the vertices v vertices v i and w and w i . Assign the label 2 to the vertex v1 . Then we assign the label 3 to all the vertices of w of w i (1 i n 1). Assign the label 1 to the vertices v 2 , v3 , v4 ,... We consider the vertices x i and y and y i . Assign the label to the vertices x i (1 i n 1) is same as assign the label to the vertices vi (1 i n 1) and assign the label to the vertices yi (1 i n 1) is same as assign the label to the vertices w i (1 i n 1). Next we we move move ′ ′′ ′ ′ ′ to the vertices v i and vi . Assign the label 2 to the vertices vertices v 1 , v2 , v3 , and assign the label ′′ ′′ ′′ 3 to the vertices v1 , v2 , v3 , . Consider the vertices x i′ and x and x i′′ . Assign Assign the label label 1 to all the ′ ′′ vertices of x of x i (1 i n 1). Assign the label 2 to the vertex x 1 . Then we assign the label 3 to the vertices x vertices x2′′ , x3′′ , x4′′ , . Now we assign the label 2 to all the vertices of w of w i′ (1 i n 1) and assign the label 3 to all the vertices of w of w i′′ (1 i n 1). Now we move to the vertices vertices y i′ and y and y i′′ . Assign the label 1 to the vertices y 1′ , y2′ , y3′ , and we assign the label 2 to the vertices vertices ′′ ′′ ′′ ′ ′′ y1 , y2 , y3 , . Next we move to the vertices u i and u and u i . Assign the label 1,3 to the vertices vertices u 1′ and u1′′ respectively respectively.. Assign Assign the label 1 to the vertices vertices u2′ i for all the values of i = 1, 2, 3, and assign the label 2 to the vertices u 2i+1 for i for i = 1, 2, 3, then we assign the label 2 to the ′′ ′′ ′′ vertices u 2 , u3 , u4 , . The vertex and edge conditions are

⊙

≤ ≤ − }

⊙

{



≤ ≤ } {

{

≤ ≤ − }

···

≡

≤ ≤ −

···

≤ ≤ −

≤ ≤ − ···

···

···

vf (1) = v = v f (2) = v = v f (3) =

≡

≤ ≤ − ···

···

≤ ≤ − ···

···

≤ ≤ −

≤ ≤ −

≤ ≤ −



≤ ≤ } {

15 15n n

− 12

3

···

128


and ef (0) = 11n 11n

− 9,

ef (1) = 11n 11n

− 10 10..

References

Electronic Journal of Combinatorics , [1] J.A.Gallian, A Dynamic Dynamic survey of graph labeling, The Electronic 19 (2016) 19 (2016) #Ds6. [2] F.Harary, F.Harary, Graph Theory , Addision wesley, New Delhi (1969). [3] R.Ponraj, M.Maria Adaickalam Adaickalam and R.Kala, k-difference k -difference cordial labeling of graphs, International Journal of Mathematical Combinatorics , 2(2016), 2(2016), 121-131. [4] R.Ponraj, M.Ma M.Maria ria Adaickalam, Adaickalam, 3-difference 3-difference cordial labeling of some union of graphs, graphs, Palestine Journal of Mathematics , 6(1)(2017), 202-210. [5] R.Ponraj, M.Maria Adaickalam, Adaickalam, 3-difference cordial labeling of cycle related graphs, Journal of Algorithms and Computation , 47(2016), 47(2016), 1-10. [6] R.Ponraj, M.Maria Adaickalam, Adaickalam, 3-difference cordiality of some graphs, Palestine Journal of Mathematics , 2(2017), 2(2017), 141-148. [7] R.Ponraj, R.Ponraj, M.M M.Maria aria Adaick Adaickalam, R.Kala, 3-difference 3-difference cordiality cordiality of corona of double double alteralternate snake graphs, Bulletin of the International Mathematical Virtual Institute , 8(2018), 245-258.


Graph Operations on Zero-Divisor Graph of Posets N.Hosseinzadeh (Department of Mathematics, Islamic Azad University, Dezful Branch, Dezful, Iran) E-mail: [email protected]

Abstract: Abstract: We know that some large graphs can be constructed from some smaller graphs by using graphs operations. operations. Many Many properties properties of such such large graphs are closely related related to those of the correspond corresponding ing smaller smaller ones. ones. In this paper we investi investigate gate some operations operations of zero-divisor graph of posets.

Key Words: Words: Poset, zero-divisor graph, graph operation. AMS(2010): AMS(2010): 06A11,05C25. §1. Introduction

In [2], Beck, for the first time, studied zero-diviso zero-divisorr graphs of the commutati commutative ve rings. Later, Later, D. F. Anderson and Livingston investigated nonzero zero-divisor graphs of the rings (see [1]). Some researchers also studied the zero-divisor graph of the commutative commutative rings. Subsequently, Subsequently, others extended the study to the commutative semigroups with zero. These can be seen in [3, 5, 7, 8].

≤

Assume (P, (P, ) is a poset (i.e., P is P is a partially partially ordered set) with the least element element 0. For every x, every x, y P , P , defined of L(x, y ) = z P z x and a nd z y . x is a zero-divisor element of P if l(x, y ) = 0, for some 0 = y P . P . Γ(P Γ(P )) is the zero-divisor graph of poset P , P , where the its vertex set consists of nonzero zero-divisors elements of P and P and x is adjacent to y if only if L(x, y ) = 0 . In this paper, P paper, P denotes denotes a poset with the least element element 0 and Z ( Z (P ) P ) is nonzero zero-divisor elements of P . P . The zero-divisor zero-divisor graph graph is undirecte undirected d graph with vertices vertices Z (S ) such that for every distinct x, y Z (S ), ), x and y are adjacent if only if L(x, y ) = 0 . Throug Throughou houtt this this paper, G always denotes a zero-divisor graph which is a simple graph (i.e., undirected graph without loops and multiple) and the set vertices of G of G show show V V ((G) and the set edges of G denotes G denotes E (G). The degree degree of vertex vertex x is the number of edges of G of G intersecting x. N ( N (x), which is the set of vertices adjacent to vertex x, is called the neighborhood of vertex x. If n is a (finite or infinite) natural number, then an n-partite n -partite graph is a graph, which is a set of vertices that can be partition partitioned ed into subsets, each of which which edges connects connects vertices vertices of two different different sets. A complete n complete n-- partite graph is a n n - partite graph such that every vertex is adjacent to the vertices which are in a different part. A graph H graph H is is a subgraph of G if G if V ( V (H ) V ( V (G) and E and E (H ) E (G). H is H is called an induced subgraph of G of G if for every x, y V ( V (H ), ), x, y E ( E (G). A subgraph subgraph H H of G is G is called a clique if H if H is is a complete graph. The clique number ω( ω (G) of G of G is is the least upper

∈

 ∈

{ ∈ | ≤

≤ }

{}

∈

{ }

∈



⊆ { } ∈

⊆

130

N.Hosseinzadeh

bound of the cliques sizes of G of G.. Many large graphs can be constructed by expanding small graphs, thus it is important to know which properties of small graphs can be transfered to the expanded ones, for example Wang in [6] proved that the lexicographof vertex transitive graphs is also vertex transitive as well well as the lexicographi lexicographicc product of edge transitiv transitivee graphs. Specapan Specapan in [9] found the fewest fewest number of vertices for Cartesian product of two graphs whose removal from the graph results in a disconnected or trivial graph. Motivated by these, we consider five kinds of graph products as the expander graphs which is described below and we can verify if regard the product of them can be regarded as a Cayley graph of the semigroup which is made by their product underlying semigroup and if the answer is positive does it inherit Col-Aut-vertex property of from the preceden precedents. ts. Let Γ = (V, ( V, E ) be a simple graph, where V is V is the set of vertices and E is E is the set of edges of G. G . An edge joins the vertex u to the vertex v is denoted by (u, ( u, v ). In [10], the authors described the following definition:

graph G is called a compact graph if G does not contain isolated Definition 1.1 A simple graph vertices and for each pair x and y of non-adjacent vertices of G, there exists a vertex z with N ( N (x) N ( N (y ) N ( N (z ).

∪

⊆

Definition 1.2 Let Γ Γ 1 = (V 1 , E 1 ) and Γ 2 = (V 2 , E 2 ) be two graphs. Γ = (V, ( V, E ), the product of them is a graph with vertex set V = V 1 V 2 , and two vertices (u ( u1 , u2 ) is adjacent to (v ( v1 , v2 ) in Γ if one of the relevant conditions happen depending on the product.

×

(1) Cartesian product. u product. u 1 is adjacent to v 1 in Γ 1 and u 2 = v = v2 or u 1 = v = v 1 and u 2 is adjacent to v2 in Γ Γ 2 ; (2) Tensor product. u1 is adjacent to v1 in Γ1 and u u 2 is adjacent to v 2 in Γ Γ 2 ; (3) Strong Strong product. product. u1 is adjacent to v1 in Γ1 and u u 2 = v 2 or u1 = v 1 and u2 is adjacent to v2 in Γ Γ 2 or u u 1 is adjacent to v 1 in Γ1 and u2 is adjacent to v 2 in Γ2 ; (4) Lexicographic. u 1 is adjacent to v1 in Γ1 or u1 = v = v 1 and u u 2 is adjacent to v2 in Γ2 ; (5) Co-normal product. u 1 is adjacent to v 1 in Γ Γ 1 or u u 2 is adjacent to v 2 in Γ2 ; (6) Modular product. u product. u 1 is adjacent to v 1 in Γ Γ 1 and u u 2 is adjacent to v 2 in Γ Γ 2 or u u 1 is not adjacent to v 1 in Γ Γ 1 and u2 is not also adjacent to v 2 in Γ2 . §2. Preliminary Notes

In this section, section, we recall recall some lemmas lemmas and definitions. definitions. from Dancheny Dancheny Lu and Tongsue Tongsue We in [10], the authors described following definition.

graph G is called a compact graph if G does not contain isolated Definition 2.1 A simple graph vertices and for each pair x and y of non-adjacent vertices of G, there exists a vertex z with N ( N (x) N ( N (y ) N ( N (z ).

∪

⊆

It has been showed the following theorem in [10].

graph G is the zero-divisor graph of a poset if and only if G is a Theorem 2.2 A simple graph

131

Graph Operations on Zero-Divisor Graph of Posets

compact graph. §3. Cartesian Product

Through this section, we assume that P and Q and Q are are two posets with the least element 0. Assume G and H and H are are in zero-divisors graphs of P of P and Q and Q,, respectively. N ( N (x) and N and N ((a) are neighborhoods in G in G and H , H , respectively, where x V ( V (P ) P ) and a and a V ( V (Q).

∈

∈

Then Theorem 3.1 Let Γ be the Cartesian product of two zero-divisor graph of G and H . Then N ( N (x, a) = (N ( N (x) a ) ( x N ( N (a)), for any (x, ( x, r) V ( V (G H ).

×{ } ∪ { }× ∈ × Proof Let (s, (s, r ) ∈ N ( N (x, a). Therefore, (s, ( s, r) is adjacent to (x, ( x, a). Thus, s Thus, s is adjacent to x in G in G and r = a = a or or s = x = x and r is adjacent to a in H . H . Hence, s Hence, s ∈ N ( N (x) and r and r = a = a or or s = x = x and r ∈ N ( N (a). It can be concluded that to N ( N (x, a) = (N ( N (x) × {a}) ∪ ({x} × N ( N (a)).

compact graphs. Then Γ the cartesian product of them is Theorem 3.2 Let G and H be two compact not a compact graph. Proof Let (x, (x, a) and (y, (y, b) be two arbitrary vertices not being adjacent adjacent of the graph Γ, where (x, a) = (y, ( y, b). Therefore, x Therefore, x and and y y are are not adjacent in G in G or or a a = b in b in H H and x and x = y in y in G G or or a, a, b are not adjacent in H in H .. Assume that there exists (z, ( z, c) V (Γ) V (Γ) such that N that N ((x, a) N ( N (y, b) N ( N (z, c). That is,





∈

(N ( N (x)

× {a}) ∪ ({x} × ⊆

 ∪

⊆

N ( N (a))

∪ (N ( N (y ) × {b}) ∪ ({y } × N ( N (b)) N ( N (z ) × {c}) ∪ ({z } × N ( N (c)). )).

Assume that (m, (m, a), (n, a) (N ( N (x) a ) such that (m, (m, a) N ( N (z ) c and (n, (n, a) 2 z N ( N (c). Then Then,, m N ( N (z ), a = c, a N ( N (c). Henc Hence, e, ac = 0 and c = 0. That is is a contradiction. Therefore, N ( N (x) a has intersection only one of N of N ((z ) c and z N ( N (c). Similary, we get this subject for ( x N ( N (a)), )), (N ( N (y ) b ) and ( y N ( N (b)).

{ }×

∈

∈

× { } ∈

∈

× { }

∈

×{ } × { } { } × { }× ×{ } { }× Now, suppose N ( N (x) × {a} ⊆ N ( N (z ) × {c}( i.e., a = c, N ( N (x) ⊆ N ( N (z )). )). If { {x} × N ( N (a) ⊆ 2 N ( N (z ) × {c}, we have N have N ((a) = {c}. Hence, ac Hence, ac = = 0. On the other hand a = a = c c,, then c then c = 0. That is a contradiction. Therefore, {x} × N ( N (a) ⊆ {z } × N ( N (c), that is x is x = = z z and N ( N (a) ⊆ N ( N (c). Then, N ( N (x) = N ( N (z ) and N and N ((a) = N ( N (c).

× { } ⊆ N ( N (z ) × {c}.

Suppose N ( N (y ) b, c, N ( N (a) = N ( N (b) = N = N ((c).

⊆ N ( N (z ) = N ( N (x), b = c. c .

Thus, Thus, N N ((y )

Hence, Hence, a = b =

If y N ( N (b) N ( N (z ) c , y N ( N (z ) and N and N ((b) = c = c.. Then, bc Then, bc = = c c 2 = 0. That is a contract. Therefore, y N ( N (b) c N ( N (z ). We get y get y = = z and N ( N (b) N ( N (c). It leads to a = a = b b = = c c and and x = y = y = = z z.. That is a contradiction.

{ } × ⊆ × { } ∈ { } × ⊆ { } ×

⊆

Corollary 3.2 Let G and H H be two compact graphs of two poset. Then, the cartesian product of them is not a graph of a poset.

Proof Referring Referring to the theorem above above and [10], it is clear. clear.

132

N.Hosseinzadeh

§4. Tensor Product

Through this section, we assume that G and H and H are are two zero-divisor graphs of poset P and Q and Q with the least element 0, respectively. Theorem 4 Theorem 4..1 Γ is the tensor product of the graphs G and H . Then N ( N (x, a) = (N ( N ((x) for any (x, ( x, a) V ( V (G H ).

∈

×

× N ( N (a))

Proof Assume (s, (s, r ) N ( N (x, a). Then, (s, (s, r) is adjacent to (x, ( x, a). By Definition 1. 1 .2, s 2, s and x are adjacent and r and a are adjacent adjacent too. Therefore, Therefore, s N ( N (x) and r N ( N (a). It leads leads to to N ( N (x, a) = N ( N (x) N ( N (a).

∈

∈

×

∈

§5. Strong Product

Through this section, we assume that H and K and K are are two zero-divisor graphs of poset P and Q and Q with the least element 0, respectively. By Definition 1. 1 .2, Theorems 3. 3 .1 and 4. 4.1, we conclude the following theorems.

posets. Then, Then, if Theorem 5.1 Γ the strong product of two zero-divisors graphs G and H of posets. runs for any (x, a) V (Γ) V (Γ), N ( N (x, a) = (N ( N ((x) a ) ( x N ( N (a)) (N ( N (x) N ( N (a))

∈

×{ } ∪ { }× ∪ × Proof By Definition 1. 1 .2, for any (r, ( r, s) ∈ N ( N (x, a), where (x, (x, a) ∈ V (Γ), V (Γ), r is adjacent to x

inG inG and s = or r = x = x and and s is adjacent to a in H or r or r is adjacent to x in G and s = a a or G and s is adjacent to aiH to aiH .. Therefore, N Therefore, N ((x, a) = (N ( N (x) a ) ( x N ( N (a)) (N ( N (x) N ( N (a)).

×{ } ∪ { }×

∪

×

§6. Co-normal Product

Theorem 6.1 Γ is the co-normal product of two graphs G and H of two the posets of P and Q, respectively. Then for any (x, ( x, a) V (Γ) V (Γ), N ( N (x, a) = (N ( N (x) V ( V (H )) )) (V ( V (H ) N ( N (a)).

∈

×

∪

×

Proof By Definition 1. 1 .2, if ((s, ((s, r) is adjacent to (x, ( x, a), s ), s and x and x are are adjacent in G in G or or r, a are adjacent in H in H .. Thus, N Thus, N ((x, a) = (N ( N ((x) V ( V (H )) )) (V ( V (H ) N ( N (a)).

×

∪

×

co-normal mal prod product uct of two comp compact act graphs graphs G and H , then Γ is a Theorem 6.2 If Γ is the co-nor compact graph. Proof Let (x, (x, a) and (y, (y, b) not be in Γ and (x, ( x, a) = (y, b). By referri referring ng the virtue virtue of Definition 1. 1.2, we get x get x and a ndyy are not adjacent in G and a and b are not adjacent in H . Then there exist z exist z G and s H such H such that N that N ((x) N ( N (y ) N ( N (z ) and N and N ((a) N ( N (b) N ( N (c). Hence,



∈

N ( N (x, a)

∈

∪ N ( N (y, b)

∪

= (N ( N (x)

⊆

∪

⊆

× N ( N (a)) ∪ (N ( N (y ) × N ( N (b) ⊆ (N ( N (z ) × N ( N (c)) ∪ (N ( N (z ) × N ( N (c)) = N ( N (z ) × N ( N (c)

Now, we get the following corollary.

133

Graph Operations on Zero-Divisor Graph of Posets

Corollary 6.3 The co-product of two zero-divisor graphs of posets is a zero-divisor graph of a poset.

Proof By the above theorem theorem and [10], it is clear. clear. §7. Lexicographic Product

Theorem 7 Theorem 7..1 Γ the lexicographic product of two zero-divisor graphs G and H H of the two posets respectively. Then, N ( P and Q Q , respectively. N (x, a) = (N ( N (x) V ( V (H )) )) ( x N ( N (a)), for any (x, ( x, a) V (Γ) V (Γ).

×

∪ { }×

∈

∈

Proof By Definition 1. 1 .2, assume (s, (s, r ) N ( N (x, a). Therefore, s Therefore, s and x are adjacent in G or s = x = x in in G G and and r r and and a a are are adjacent in H in H .. Therefore, N Therefore, N ((x, a) = (N ( N (x) V ( V (H )) )) ( x N ( N (a)), for any (x, (x, a) V ( V (γ ). ).

×

∈

∪ { }×

§8. Modular Product

Theorem 8.1 Γ the Modular product of two zero-divisor graphs G and H of the two posets P P and Q respectively. ely. Then, N ( Q respectiv N (x, a) = (N ( N ((x) N ( N (a)) (N c (x) N c (a)), for any (x, ( x, a) V (Γ) V (Γ).

× ∪ Proof By Definition 1. 1.2, assume (s, (s, r) ∈ N ( N (x, a).

×

∈

Therefore, Therefore, s and x are adjacent in G while r and a are adjacent in H or s and x are not adjacent in G whereas r and a are not adjacent in H in H .. Thus, N Thus, N ((x, a) = (N ( N ((x) N ( N (a)) (N c (x) N c (a)).

×

∪

×

References [1] D. F. Anderson, Anderson, P. S. Livingston Livingston,The ,The zero-Divisor zero-Divisor Graph of a Commutati Commutative ve Ring, J. Algebra , 159 (1991), 500-514. [2] I. Beck, Coloring of Commutative Commutative Rings, J. Algebra , 116 (1988), 208-226. [3] F. R. DeMeyer, DeMeyer, L. DeMeyer, DeMeyer, Zero-Divisor Zero-Divisor Graphs of Semigroup Semigroups, s, J. Algebra , 283 (2005), 190-198. [4] Dancheny Lu and Tongsue Tongsue We, We, The zero-divisor graphs of posets p osets and application to semigroups, Graphs and Combinatorics , (2010) 26, 793-804. [5] F. R. DeMeyer, T. McKenzie McKenzie and K. Schneider, Schneider, The Zero-Divisor Graphs of a Commutative Commutative Semigroup, Semigroup Forum , 65 (2002), 206-214. [6] F. Li, W. Wang, Z. Xu and H. Zhao, Some results on the lexicograp lexicographic hic product of vertexvertex Applied Math. Letters , 24(2011), 1924-1926. transitive graphs, Applied [7] S. E. Wright, Lengths of paths and cyclies in zero-divisor graphs and digraphs of semigroups, Comm. Algebra , 35 (2007), 1987-1991. [8] T. Wu and F. Cheng, Cheng, The structure structure of zero-divisor zero-divisor semigroup semigroup with graph K n K 2 , Semigroup Forum , 76(2008), 330-340. [9] S. Spacapan, Connectivit Connectivity y of Cartesian product of graphs, graphs, Applied Mathematica Letters , Vol. 21, Issue 7, (2008), (2008), 682-685. [10] Dancheny Lu and Tongsue Tongsue We, We, The zero-divisor graphs of posets p osets and application to semigroups, Graphs and Combinatorics , (2010) 26, 793-804.

◦

134

International International Journal of Mathematical Mathematical Combinatorics Combinatorics

The tragedy of the world is that those who are imaginative have but slight experience, and those who are experienced have feeble imaginations. By Alfred North Whitehead, A British philosopher and mathematician

Author Information Submission: Papers only in electronic electronic form are considered for possible possible publication. publication. Papers prepared in formats, viz., .tex, .dvi, .pdf, or.ps may be submitted electronically to one member of the Editorial Board for consideration in both in International Journal of Mathematical Combinatorics and Mathematica Mathematicall Combinatori Combinatorics cs (Interna (International tional Book Series) Series). An effort is made to publish a paper duly recommended by a referee within a period of 3 4 months. months. Articles Articles received received are immediately immediately put the referees/mem referees/members bers of the Editorial Board for their their opinion opinion who genera generally lly pass pass on the same same in six week’ week’ss time time or less. less. In case of clear clear recommend recommendation ation for publication publication,, the paper is accommodated accommodated in an issue to appear next. Each submitted paper is not returned, hence we advise the authors to keep a copy of their submitted papers for further processing.

−

Abstract: Authors Authors are requested requested to provide an abstract of not more than 250 words, words, latest Mathematics Subject Classification of the American Mathematical Society, Keywords and phrases. phrases. Statemen Statements ts of Lemm Lemmas, as, Propositions Propositions and Theorems should be set in italics italics and references should be arranged in alphabetical order by the surname of the first author in the following style: Books [4]Linfan Mao, Combinatorial Geometry with Applications to Field Theory , InfoQuest Press, 2009. Algebraic topology topology:: an introduction introduction , Springer-Verlag, New York 1977. [12]W.S.Massey, Algebraic Research papers [6]Linfan Mao, Mathematics on non-mathematics - A combinatorial contribution, International J.Math. Combin. , Vol.3(2014), 1-34. [9]Kavita Srivastava, On singular H-closed extensions, Proc. Amer. Math. Soc. (to appear). Figures: Figures Figures should be drawn by TEXCAD in text directly directly,, or as EPS file. In addition, addition, all figures and tables tables should be numbered numbered and the appropriate appropriate space reserved in the text, text, with the insertion point clearly indicated. Copyright: It is assumed that the submitted submitted manuscript manuscript has not been published published and will not be simultan simultaneously eously submitte submitted d or published published elsewhere. elsewhere. By submitting submitting a manuscrip manuscript, t, the authors agree that the copyright for their articles is transferred to the publisher, if and when, the paper is accepted for publication. The publisher cannot take the responsibility of any loss of manuscript. manuscript. Therefore, Therefore, authors authors are requested requested to maintain maintain a copy at their end. Proofs: One set of galley proofs of a paper will be sent to the author submitting submitting the paper, unless requested otherwise, without the original manuscript, for corrections after the paper is accepted for publication on the basis of the recommendation of referees. Corrections should be restricted to typesetting errors. Authors are advised to check their proofs very carefully before return.

June June 20 2018 18

Contents Ricci Soliton and Conformal Ricci Soliton in Lorentzian β -Kenmotsu -Kenmotsu Manifold By Ta Tamali alika Du Dutta an and Ar Arindam Bh Bhattacharyy ryya . .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. . .. .. .. 01 Some Properties of Conformal β β -Change By H.S.Shukla and Neelam Mishra ................... ................................... 13 Equitable Coloring on Triple Star Graph Families By K.Praveena and M.Venkatachalam .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . 24 On the Tangent Vector Fields of Striction Curves Along the Involute and Bertrandian Frenet Ruled Surfaces By S ¸ eyda Kılı¸ Kılı ¸co˘ co˘glu, glu, S¨ uley u leyma man n S¸enyu ¸ enyurt rt and and Abdus Abdussa same mett C ¸ alı¸ alı¸skan. s kan. . . . . . . . . . . . . . . . . . . . . . . . 33 On the Leap Zagreb Indices of Generalized xyz-Point-Line Transformation Graphs T xyz (G) when z = 1 By B. Ba Basavanagoud an and Chi Chitra E.. E.. .. .. .. .. .. .. .. . .. .. .. 44 A Generalization on Product Degree Distance of Strong Product of Graphs By K. K.Pattabiraman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 67 Semifull Line (Block) Signed Graphs By V. Lokesha, P. S. Hemavathi and S. Vijay............................................80 Accurate Independent Domination in Graphs By B. Basavanagoud and Sujata Timmanaikar...........................................87 On r-Dynamic Coloring of the Triple Star Graph Families By T.Deepa and M. Venkatachalam ................... ..................................97 (1,N)-Arithmetic Labelling of Ladder and Subdivision of Ladder By V.Ramachandran...................................................................114 3-Difference Cordial Labeling of Corona Related Graphs By R. R.Ponraj and M. M.Maria Ad Adaickalam .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 12 122 Graph Operations on Zero-Divisor Graph of Posets By N.Hosseinzadeh ....................................................................129 Mathematical Combinatorics (Book Series)

Mathematical Combinatorics (International Book Series), vol. 2 / 2018

Recommend Documents