Some Topological Indices of Edge Corona of Two Graphs

Full text

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Some Topological Indices of Edge Corona of Two

Graphs

CHANDRASHEKAR ADIGA1,, MALPASHREE RAJU1, RAKSHITH BILLVA RAMANNA2 AND ANITHA NARASIMHAMURTHY3

1

Department of Studies in Mathematics, University of Mysore, Manasagangothri, Mysore−570 006, India

2

Department of Mathematics, Vidyavardhaka College of Engineering, Mysore−570002, India

3

Department of Mathematics, PES University, Bangalore−560 085, India ARTICLE INFO ABSTRACT

Article History:

Received: 13 August 2015 Accepted: 12 March 2017

Published online 30 September 2019 Academic Editor: Hassan Yousefi−Azari

In this paper, we compute the Wiener index, first Zagreb index, second Zagreb index, degree distance index and Gutman index of edge corona of two graphs.

© 2019 University of Kashan Press. All rights reserved Keywords:

Edge corona Wiener index Zagreb index Degree distance index Gutman index

1.

I

NTRODUCTION

Throughout this paper we consider only simple connected graphs, i.e., connected graphs without loops and multiple edges. Let G = ( , )V E be a graph with vertex set

1 2

( ) = { , , , n}

V G v v v and edge set E G( ) = { ,e e1 2, ,em}. We denote the shortest distance between two vertices u and v in G by d u v( , ) and the degree of a vertex v in Gby d v( ). A topological index is a real number derived from the structure of a graph, which is invariant under graph isomorphism. The Wiener index W(G) of a graph G is defined as

Corresponding Author (Email address: c_adiga@hotmail.com)

DOI: 10.22052/ijmc.2017.34313.1132

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{ , } ( )

( ) = u v V G ( , ).

W G

d u v ‎‎ The Wiener index is a classical distance based topological index. It was first introduced by H. Wiener [26] with an application to chemistry. The Wiener index has been extensively studied by many chemists and mathematicians. For more details about Wiener index the reader may refer to [5, 6, 12, 16, 20]. The first and second Zagreb indices of a graph denoted by M G1( ) and M G2( ), respectively, are degree based topological indices introduced by Gutman and Trinajstić [15]. These two indices are defined as

2 1( ) e uv E G= ( ) ( ( ) ( )) v V G( ) ( ) ,

M G 

d u d v 

d v

and M G2( )

e uv E G  ( )d u d v( ) ( ). More information about Zagreb indices can be found in [13, 14, 19]. The degree distance, DD(G), and Gutman index Gut(G) of a graph G are defined, respectively, as

{ , } ( )

{ , } ( )

( ) ( ( ) ( )) ( , ),

( ) ( ) ( ) ( , ).

G G G

u v V G

G G G

u v V G

DD G d u d v d u v Gut G d u d v d u v

 

The degree distance was introduced independently by A. A. Dobrynin, A. A. Kochetova [7] and Gutman [11]. The Gutman index was introduced by Gutman [11] which was earlier known as modified Schultz index. Studies on degree distance and Gutman index can be found in [2, 3, 8, 21, 25] and the references cited therein.

The corona [9] of two graphs G and His the graph obtained by taking one copy of G, |V(G)| copies of H and joining i-th vertex of G to every vertex in the i-th copy of H. The edge corona [17] of two graphs G and H denoted by GH is obtained by taking one copy of G and |E(G)| copies of H and joining end vertices of i-th edge of G to every vertex in the i-th copy of H. Various researchers studied the topological indices of the corona of two graphs, for example see [23, 27]. Motivated by these works, in this paper we compute some topological indices of edge corona of two graphs. In Section 2, we derive formulas for the Wiener index and Zagreb indices of edge corona of two graphs. As special cases of these formulas we express Wiener index and Zagreb indices of edge corona PnH, CnH and KnH, where H is an arbitrary graph, in terms of vertices and edges. In Section 3, we compute the degree distance and Gutman index of edge corona of two graphs and obtain formula in terms of vertices and edges for degree distance and Gutman index of edge corona of two graphs G and H, when G= Cn , Pn or Kn and H= Cm , Pm , Km or Km.

2.

W

IENER

I

NDEX

,

F

IRST

Z

AGREB

I

NDEX AND

S

ECOND

Z

AGREB

I

NDEX OF

E

DGE

C

ORONA OF

T

WO

G

RAPHS

Let G1 and G2 be graphs with vertex sets V(G1)={v1,v2,…,vn1}, V(G2)={u1 ,u2,…,un2},and

edge sets E(G1)={e1,e2,…,em1} and E(G2)={ ' 1

e ,e2',…, 2 '

m

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of G2 be denoted by V(G2)={ui1,ui2,…,uin2}. In this section, we compute the Wiener index

and Zagreb indices of edge corona of two graphs G1 and G2.

Let e = uv and f = xy be two distinct edges in G. The distance dG(e,f) between two edges e and f is given by min{dG(x,u), dG(x,v), dG(y,u), dG(y,v)}. The edge-Wiener index

( )

e

W G [4] of a graph G, is the sum of all distances between unordered pairs of vertices in the line graph L(G). Equivalently, edge-Wiener index of G is the Wiener index of the line graph L(G) of G i.e.,

{ , } ( )

( ) = ( ( )) = ( { ( , ), ( , ), ( , ), ( , )} 1).

e xy uv E G G G G G

W G W L G

min d x u d x v d y u d y v 

Note that in literature different definitions of the edge-Wiener index and distance between two edges are proposed; for example see [18]. The vertex PI index [1] of a graph G is defined as

= ( )

( ) = [ i( | ) i( | )],

i l m e l e m

e v v E G

PI G

n v G n v G

where ne=uv(u|G) is the number of vertices in G that are closer to u than v in G. It may be noted that if G is a bipartite graph, then PI(G)=|V(G)||E(G)|.

We need the following two lemmas to prove our main results. As the proofs directly follow from the definition of the edge corona, we omit the details.

Lemma 2.1 We have

1.

1 2( ) = ( 2 1) 1( )

G G i G i

d v n  d v , v V Gi  ( )1 .

2. ( )= ( ) 2

2 2

1G ij G j

G u d u

d , uij V Gi( 2).

Lemma 2.2 We have

1.

1 2( , ) = 1( , ), , ( ).1

G G i j G i j i j

d  v v d v v v v V G

2.

1 2

2

2

1, ( )

( , ) =

2, ( )

j k G G ij ik

j k

u u E G d u u

u u E G

 

 , u uij, ik V Gi( 2).

3.

1 2( , ) = 1( , ) 2,

G G ij km G i k

d u u d e e  uij V Gi( 2) andukmV Gk( 2).

4.

1 1 1

1 2

1 1 1 1

2 1

2 1

( , ) 1, ( , ) = ( , ),

( ) , ( ), ( , ) =

( ( , ) ( , ) 1) / 2, ( , ) ( , ),

( ) , ( ),

G l k G l k G m k

ij i k

G G ij k

G l k G m k G l k G m k

ij i k

d v v d v v d v v

u V G v V G d u v

d v v d v v d v v d v v u V G v V G

 

 

where v vl m =ei.

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2 2 2

1 2 1 2 1 1 1 1 2 1 1 1 2

( ) = ( ) ( ) ( ( ) ( )) ( 1) 1 .

2 2

e

n n

W G G W G n W G  DD G PI G m n  m    n  m m

 

Proof. We have

1 2 1 2 3 4

( ) = ,

W G G A A A A (2.1) where 1 2 1 1 2 1 2 1 2

1 2 2

1 2

1 2 1

1 { , } ( )

2 ( ) { , } ( )

3 { , } ( ) ( ), ( )

4 ( ) ( ), ( )

= ( , ),

= ( , ),

= ( , ),

= ( , ).

i j

i ij ik i

i k ij i km k

i ij i k

G G i j

v v V G

G G ij ik

e E G u u V G

G G ij km

e e E G u V G u V G

G G ij k

e E G u V G v V G

A

d

v v

A

d

u u

A

d

u u

A

d

u u

            

We now compute Ai for i = 1,2,3,4.By Lemma 2.2, we have

1 2 1

1 1

1 2

1 2

1 2 2

1 2 2

1 { , } ( ) { , } ( ) 1

2 ( ) { , } ( )

( ) ( ) (( )

( ) ( ) ( )

2

= ( , ) ( , ) ( ), (2.2)

= ( , ),

= 2 1

2 1

= (

i j i j

i ij ik i

i ij ik i ij ik i

i ij ik i ij ik i

G G i j G i j

v v V G v v V G

G G ij ik e E G u u V G

e E G u u E G u u E G

e E G u u V G u u E G

A d v v d v v W G

A d u u

n                 

1 2 2

( )

1 2 2 2

( 1) )

= ( ( 1) ), (2.3)

i

e E G n m

m n n m

  

 

and

1 2

1 2 2

1

1 2 2

1 1

3 { , } ( ) ( ), ( )

{ , } ( ) ( ), ( )

2

2 { , } ( )

2

2 1 1 1

= ( , ),

= ( ( , ) 2),

= ( ( , ) 2)

= ( ( ) ( 1) / 2). (2.4)

i k ij i km k

i k ij i km k

i k

G G ij km e e E G u V G u V G

G i k e e E G u V G u V G

G i k e e E G

e

A d u u

d e e

n d e e

n W G m m

           

We have

A

4 

e E Gi ( 1)

uijV Gi( 2),v V Gk ( 1)

d

G G1 2(

u u

ij, k).

Using Lemma 2.2, the above equation can be rewritten as follows:

1 1

1 1 1 1

1 1

1 1 1

1 1

1 1

4 2 ( ) ( ), ( , ) ( , )

( ), ( , ) ( , )

2

= ( ) ( )

= ( ( ( , ) ( , ) 2) / 2

( ( , ) ( , ) 1) / 2

( ( ( , ) ( , ))

l m k G m k G l k

k G m k G l k

i l m k

G l k G m k v v E G v V G d v v d v v

G l k G m k v V G d v v d v v

G l k G m k e v v E G v V G

A n d v v d v v

d v v d v v

n d v v d v v

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1

1

1 1 1

1

1 1

1

1 1 1

= ( )

2 1 1 1

( ) ( ) ( )

2 1 1 1

( ) ( )

2 1 1 1

2 ( ( | ) ( | ))) / 2

( ( , ) 2 ( )) / 2

= ( ( ) ( , ) 2 ( )) / 2

= ( ( ( ) ( )) ( , ) 2 ( )

i i

i l m

j i j k

i k

e l e m

e v v E G

G i k v V G v v V G v V G

i G i k v V G v V G

i k G i k

n n v G n v G

n d v v n m PI G

n d v d v v n m PI G

n d v d v d v v n m PI G

  

 

  

  

 

  

1

{ , } ( )

2 1 1 1 1

) / 2

= ( ( ) 2 ( )) / 2. (2.5)

i k v v V G

n DD G n m PI G

 

Using (2.2), (2.3), (2.4) and (2.5) in (2.1) the result follows.

Corollary 2.4 Let G1 be a bipartite graph. Then the Wiener index of G G12 is given by

2

1 2 1 2 1 2 1 1 1 1 2 2 1 1 1 2

( ) = ( ) e( ) ( ( ) ) / 2 ( 1) / 2 1 .

W G G W G n W G n DD G n m m n n m    n m m

We need the following lemmas for future references.

Lemma 2.5 [22] Let Pn and Cn denote the path and cycle on n vertices, respectively. Then

2

( n) = ( 1) / 6,

W P n n 

3

2 / 8, ( ) =

( 1) / 8, .

n

n if n is even W C

n n if n is odd 



Lemma 2.6 [8, 24] Let Pn and Cn denote the path and cycle on n vertices, respectively. Then

DD P( n) = (n n1)(2n1) / 3,

3

2

/ 2, ( ) =

( 1) / 2, .

n

n if n is even DD C

n n if n is odd

 

 

It can be easily verified that W K( n) =n n( 1) / 2, 2

( n) = ( 1)

DD K n n ,W Ke( n) =n n( 1)(n23n2) / 4,PI K( n) = n n( 1),PI P( n) = n n( 1)and PI C( 2n1)2n (2n1).

Using these facts and also by applying above two lemmas in Theorem 2.3, we obtain the following corollary.

Corollary 2.7 Let G be a graph with n1 vertices and m1 edges. Then

1. 2 2 2

1 1 1 1

( n ) = ( 1)(( 1) ( 1) 6 6 ) / 6.

W P G n n  n  n  n n  m

2. 2 2 2 2

2 1 1 1

( n ) = (( 1) ( 1)2 2 ).

W C G n n n  n  n n n  m

3. 2 2 2 2

2 1 1 1 1 1 1

( n ) = ( 1/ 2)(( 1) (3 4 1) 2 2 ).

W C G n n  n  n  n  n n  m

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Theorem 2.8 The first Zagreb index of G G12 is given by

2

1( 1 2) = ( 2 1) 1( )1 1( 1( 2) 4 2 8 2).

M G G n  M G m M G  n  m

Proof. By Lemma 2.1, we have

1 2 1 2

1 2 1 2

1 1 2

1 2

1 2

2 2

1 2

2

1 1 2 ( )

2 2

( ) ( ) ( )

2 2 2

2 ( ) (1) ( )

2 2

2 1 1 ( ) ( )

( ) = ( )

= ( ) ( )

= ( 1) ( ) ( ( ) 2)

= ( 1) ( ) ( ( ) 4 ( ) 4)

=

i i ij i

i i ij i

i ij i

G G x V G G

G G i G G ij

v V G e E G u V G

G i G j

v V G e E G u V G

G j G j

e E G u V G

M G G d x

d v d u

n d v d u

n M G d u d u

                     

1 2

2 1 1 ( ) 1 2 2 2

2

2 1 1 1 1 2 1 2 2 1

( 1) ( ) ( ( ) 8 4 )

= ( 1) ( ) ( ) 8 4 .

i e E G

n M G M G m n

n M G m M G m m n m

   

   

This completes the proof.

Theorem 2.9 The second Zagreb index of G G12 is given by

2

2( 1 2) = ( 2 1) 2( )1 1 2( 2) 2 1 1( 2) 4 1 2 2( 2 1)( 2 2) 1( ).1

M G G n  M G m M G  m M G  m m  n  m n M G

Proof. By Lemma 2.1, we have

1 2 1 2 1 2

1 2 1 2 1 2 1 2

1 1 2

1 2 1 2 1 2

1 2

1

2 1 2 ( )

( ) ( ) ( )

= ( ) ( )

2

2 ( )

( ) = ( ) ( )

= ( ) ( ) ( ) ( )

( ( ) ( )) ( )

= ( 1)

i j i ij ik i

i l m ij i

i j

G G G G xy E G G

G G i G G j G G ij G G ik

v v E G e E G u u E G

G G l G G m G G ij e v v E G u V G

v v E G

M G G d x d y

d v d v d u d u

d v d v d u

n                      

1 1 2 2 1 2

1 1 2

1 2

1

1 1

( ) ( )

2 = ( ) ( )

2

2 2 1 ( ) 2 2 1 2 2

2 2

( ) ( )

( ( ) 2)( ( ) 2)

( 1) ( ( ) ( )) ( ( ) 2)

= ( 1) ( ) ( ( ) 2 ( ) 4 )

( 1) ( ( ) ( )) (2

i ij ik i

i l m ij i

i

G i G j

G ij G ik e E G u u E G

G l G m G ij e v v E G u V G

e E G

G l G m

d v d v

d u d u

n d v d v d u

n M G M G M G m

n d v d v m

                   

1 2 = ( ) 2

2 2 1 1 2 2 1 2 2

2 2 2 1 1

2 )

= ( 1) ( ) ( ( ) 2 ( ) 4 )

2( 1)( ) ( ).

i l m

e v v E G n

n M G m M G M G m n m n M G

   

  

Hence the proof.

Using the facts that M P1( n) = 4n6(n2), M P2( n) = 4n8 (n3),M C1( n) =

2( n)

M C =4n,M K1( n) = (n n1) ,2 M K2( n) = (n n1) / 23 , in Theorems 2.8 and 2.9, we

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Corollary 2.10 We have

1. M P P1( n m) = 4m n2 6m224mn28m10n8.

2. M C C1( n m) = 4 (n m26m1).

3. M P C1( n m) = 4m n2 6m224mn28m4n6.

4. M C P1( n m) = 4m n2 24mn10 .n

5. 2

1( n m) = (1/ 2) ( 1)( 1) ( 2 2).

M K K n n m m n

6. 2

1( n m) = ( 1)(( 1)( 1) 8 7).

M K P n n n m  m

7. 2

1( n m) = ( 1) ( 4 6).

M P K m mn m  n

8. 2 2

1( n m) = ( 1)(( 1) 6 1).

M K C n n m n m  m

9. 3 2

1( n m) = 6 9 4 .

M C K m n m n mn n

10.M P K1( nm) = 4m n2 6m212mn16m4n6.

11.M C K1( nm) = 4 (n m23m1).

12.M K K1( n m) = (n n1)((m1) (2 n 1) 2 ).m

Corollary 2.11 We have

1. 2

2( n m) = (20 32) (32 44) 28 28.

M P P n m  n m n

2. 2

2( n m) = 20 40 4 .

M C C m n mn n

3. 2

2( n m) = (20 32) (40 56) 4 8.

M P C n m  n m n

4. M C P2( n m) = 20m n2 32mn28 .n

5. M K K2( n m) = (1/ 4) (n n1)(m2(4n5)m2(n1) )(2 m1) .2

6. M K P2( n m) = (1/ 2) (n n1)(m n2 26m n2 2mn27m2n214m6n19).

7. M P K2( n m) = (1/ 2((n1)m2(7n11)m8n16))(m1) .2

8. M K C2( n m) = (1/ 2) (n n1)((m1)2n2(6m24m2)n7m210m1).

9. 2 2

2( n m) = (1/ 2) ( 7 8)( 1) .

M C K n m  m m

10.M P K2( nm) = (12n20)m2(16n28)m4n8.

11.M C K2( nm) = 12m n2 16mn4 .n

12.M K K2( nm) = (1/ 2) (n m1)(n1) ((2 m1)n3m1).

3. D

EGREE DISTANCE AND

G

UTMAN INDEX OF EDGE CORONA OF TWO GRAPHS

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( ), = ( ) ( , ) = ( , )

( ) := ( ).

G G

G w V G e uv E G

d w v d u w

C G

d w

It is easy to see that C C( 2n1) = 4n2 and C K( n) = (n n1) (2 n2) / 2.

In this section, we compute the degree distance and Gutman index of edge corona of two graphs.

Theorem 3.1 The degree distance index of G G12 is given by

1 2 2 2 1 2 2 2 1

2 2 1 1 2 2 1

2

1 1 2 1 2 2 1 1 2 1 2 1 2 2

( ) = ( 2 1) ( ) 4 ( ) ( )

(1 / 2) ( 1)(2 ( ) ( )) ( ) ( )

( ) 2 ( )( ( 1) 2) 4 ( 1).

e

DD G G m n DD G n m n W G

n n Gut G C G m n PI G

m M G m m n n m n m m m n n

    

    

        

Proof. We have

1 2 1 2 3 4

( ) = ,

DD G G A A A A

(3.1) where

1 2 1 2 1 2 1

1 2 1 2 1 2 1

2

1 2 1 2 1 2

1 2 2

1

{ , } ( )

2 ( ) { , } ( )

3 { , } ( ) ( ), ( )

= ( ( ) ( )) ( , ),

= ( ( ) ( )) ( , ),

= ( ( ) ( )) (

i j

i ij ik i

i k ij i km k

G G i G G j G G i j v v V G

G G ij G G ik G G ij ik e E G u u V G

G G ij G G km G G ij e e E G u V G u V G

A d v d v d v v

A d u d u d u u

A d u d u d u

                

1 2 1 2 1 2

1 2 1

4 ( ) ( ), ( )

, ),

= ( ( ) ( )) ( , ).

i ij i k

km

G G ij G G k G G ij k e E G u V G v V G

u A

d u d v d u v

Using Lemmas 2.1 and 2.2, A1, A2, A3 and A4 can be computed as follows:

1 2 1 2 1 2 1

1 1 1

1

1 { , } ( )

2 { , } ( )

2 1

= ( ( ) ( )) ( , )

= ( 1) ( ( ) ( )) ( , )

= ( 1) ( ). (3.2)

i j

i j

G G i G G j G G i j v v V G

G i G j G i j v v V G

A d v d v d v v

n d v d v d v v

n DD G

        

1 2 1 2 1 2

1 2 2 2 1 2 2 2 2 2 1 2

2 ( ) { , } ( )

( ) { , } ( )

( )

2 2 2 1 2

( ) ( )

= ( ( ) ( )) ( , )

= (2 ( ( ) ( ) 4)

( ( ) ( ) 4))

= (2( 1) ( ( )) 4 ( 1) ( )

i ij ik i

i j k

j k

i j

G G ij G G ik G G ij ik e E G u u V G

G j G k e E G u u V G

G j G k u u E G

G j

e E G u V G

A d u d u d u u

d u d u d u d u

n d u n n M G

                   

1 2

2 2 2 2 1 2 2

( )

2 1 2 1 2 2 1 1 2

4 )

= 4( 1) 4 ( 1) ( ) 4

4( 2) 4 ( 1) ( ). (3.3)

i e E G

m n m n n M G m

n m m m n n m M G

    

    

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1 2 1 2 1 2

1 2 2

2 1

1 2

1 1

3 { , } ( ) ( ), ( )

2 { , } ( ) ( )

2 { , } ( ) 2 2

2 2 2 1 1 1

= ( ( ) ( )) ( , )

= 2 ( ( ) 2)( ( , ) 2)

= 2 ( ( , ) 2)(2 2 )

= 4 ( )( ( ) ( 1)

i k ij i km k

i k

i k

G G ij G G km G G ij km e e E G u V G u V G

G i G i k

e e E G ui V G

G i k e e E G

e

A d u d u d u u

n d u d e e

n d e e m n

n m n W G m m

                

1 2 1 2 1 2

1 1

1 1

1 1

1

4 ( ) ( 2), ( )

2 2

( ) ( )

2 2

2 2 1 1 1

1 2

/ 2). (3.4)

= ( ( ) ( )) ( , )

= ((1 / 2) ( ( , ) ( , ))(2( )

( 1) ( ))

( )(2 ( ( | ) ( | )))

(

i ij i k

l m k

i i

G G ij G G k G G ij k e E G u V G v V G

G l k G m k v v E G v V G

G k

e l e m

A d u d v d u v

d v v d v v m n n n d v

m n n n v G n v G m n n

                 

1 1 1 1

2 1) (1 / 2) n n2( 21)

vkV G d v v( ),G(m, k)d v vG(l,k)d vG ( k).

Thus,

4 2 2 1 1 2 2 1 1

2

2 2 1 1 1 2 2

= ( )( ( ) ( )) (1/ 2) ( 1)(2 ( ) ( ))

( )(2 ) ( 1). (3.5)

A m n DD G PI G n n Gut G C G m n m n m n n

    

   

Employing (3.2), (3.3), (3.4) and (3.5) in (3.1), we get the required result.

Corollary 3.2 Let G be a bipartite graph. Then 1

1 2 2 2 1 2 2 2 1

2 2 1 2 2 1 1 1 1 2

2

1 2 2 1 1 2 1 2 1 2 2

( ) = ( 2 1) ( ) 4 ( ) ( )

( 1) ( ) ( ) ( )

2 ( )( ( 1) 2) 4 ( 1).

e

DD G G m n DD G n m n W G n n Gut G m n n m m M G

m m n n m n m m m n n

    

    

       

Theorem 3.3 The Gutman index of G G12 is given by

2 2

1 2 2 2 2 1 2 2 2 2 1

1 1 2 1 2 2 2 2 2 1

1 2 1 2 2 2 2 1 2 1 2 2

( ) = ( 1)(2 3 1) ( ) (4 8 4 ) ( )

3 ( ) ( ) ( )( 1) ( )

2 ( (2 3 1) 2 2) 2 ( ( 1) 6).

e

Gut G G n m n Gut G m n m n W G m M G m M G m n n C G

m n m n m n m m m m m m

      

    

         

Proof.

1 2 1 2 3 4

( ) = ,

Gut G G A A A A

(3.6) where

1 2 1 2 1 2 1

1 2 1 2 1 2

1 2

1 2 1 2 1 2

1 2 2

1 { , } ( )

2 ( ) { , } ( )

3 { , } ( ) ( ), ( )

= ( ) ( ) ( , ),

= ( ) ( ) ( , ),

= ( ) ( ) ( ,

i j

i i

i k i km k

G G i G G j G G i j v v V G

G G ij G G ik G G ij ik e E G u uij ik V G

G G ij G G km G G ij e e E G uij V G u V G

A d v d v d v v

A d u d u d u u

A d u d u d u u

              

1 2 1 2 1 2

1 2 1

4 ( ) ( ), ( )

),

= i k ( ) ( ) ( , ).

km

G G ij G G k G G ij k e E G uij V G u V Gi

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Using Lemmas 2.1 and 2.2, A1, A2, A3 and A4 can be computed as follows:

1 2 1 2 1 2 1

1 1 1

1

1 2 1 2 1 2

1 2

1 { , } ( )

2

2 { , } ( )

2

2 1

2 ( ) { , } ( )

{ ,

= ( ) ( ) ( , )

= ( 1) ( ) ( ) ( , )

= ( 1) ( ). (3.7)

= ( ) ( ) ( , )

= (2

j

j

i ij ik

j

G G i G G j G G i j v vi V G

G i G j G i j v vi V G

G G ij G G ik G G ij ik e E G u u V Gi

u u

A d v d v d v v

n d v d v d v v

n Gut G

A d u d u d u u

           

2 2 2 2

1 2

2 2 2 2

2 1

( ) } ( )

, ( )

2

2 1 2 2 2 2 2

( )

2 2 1 2 2

2

1 2 1 2 2 2 2

( ( ) ( ) 2( ( ) ( )) 4)

( ( ) ( ) 2( ( ) ( )) 4)

= (4 ( )) 8( 1) 4 ( 1)

( ) 2 ( ) 4 )

= (4 3 ( ) ( ) 4( 1)(

i k

j k

i

G j G k G j G k

e E G V G

G j G k G j G k

u u E G

e E G

d u d u d u d u d u d u d u d u

m M G n m n n M G M G m

m m M G M G n

                      

2 2 2

2m n )4m ). (3.8)

1 2 1 2 1 2

1 2 2

2 2 1

1 2 2

1

1 2

3 { , } ( ) ( ), ( )

{ , } ( ) ( ), ( )

{ , } ( ) ( ),

= ( ) ( ) ( , )

= ( ( ) 2)( ( ) 2)( ( , ) 2)

( ( , ) 2) (

i k ij km k

i k ij km k

i k km

G G ij G G km G G ij km e e E G u V G ui V G

G ij G km G i k

e e E G u V G ui V G

G i k

e e E G uij V G ui V

A d u d u d u u

d u d u d e e d e e

                

2 2 2

2 2 2 2 1 1 ( ) 2 2 ( ), ( ) 2 2

2 2 2 2

{ , } ( )

2 2

2 2 2 2 1 1 1

( ) ( )

2 ( ( ) ( )) 4 )

= ( ( , ) 2)(4 8 4 )

= (4 8 4 )( ( ) ( 1) / 2). (3.9)

ij i km k

i k

G ij G km G

k

G ij G km u V G u V G

G i k e e E G

e

d u d u

d u d u n

d e e m n m n m n m n W G m m

            

1 2 1 2 1 2

1 2 1

1 1 1

1 1

1 1 1 1

4 ( ) ( ), ( )

2 2 2

( ) ( )

2 2 2 1 ( ), ( , ) ( , )

2

= ( ) ( ) ( , )

= (( )( 1)( ( , ) ( , )) ( )

( )( 1)(2 ( )))

= (

i ij i k

l m k

k G m k G l k

G G ij G G k G G ij k e E G u V G v V G

G l k G m k G k v v E G v V G

G k v V G d v v d v v

A d u d v d u v

m n n d v v d v v d v

m n n m d v

m n                  

2

2)(n2 1)[2Gut G( 1)C G( 1)2m1]. (3.10)

Now, the result follows using (3.7), (3.8), (3.9) and (3.10) in (3.6).

Corollary 3.4 Let G1 be a bipartite graph. Then

2 2

1 2 2 2 2 1 2 2 2 2 1

1 1 2 1 2 2 1 2 1 2 2 2

2 1 2 1 2 2

( ) = ( 1)(2 3 1) ( ) (4 8 4 ) ( )

3 ( ) ( ) 2 ( (2 3 1)

2 2) 2 ( ( 1) 6).

e

Gut G G n m n Gut G m n m n W G m M G m M G m n m n m n

m m m m m m

      

     

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Lemma 3.5 [10] Let Pn and Cn denote the path and the cycle on n vertices, respectively. Then

2

( n) = ( 1)(2 4 3) / 3,

Gut P n n  n

3

2

/ 2, ,

( ) =

( 1) / 2, .

n

n if n is even Gut C

n n if n is odd

 

 

Using Lemmas 2.5, 2.6 and 3.5 in Theorems 3.1 and 3.3, we obtain the following corollaries.

Corollary 3.6 We have

1. 2 2 2 2

( n m) = ( 1)(2 2 16 14).

DD P P n m n m n mn  m n 

2. 2 3 2 2 2

2

( n m) = (12 16 4) (20 12 ) (8 32 ) .

DD C C m  m n  m  m n  m  m n

3. 2 2 2 2 2 2

2 1

( n m) = 2(2 1)(3 8 4 5 7 5 ).

DD C C n m n  m n mn  m  mn n  m n

4. 2 2 2 2 2

( n m) = (1/ 3)( 1)(6 3 8 2 2 48 ).

DD P C n m n  m n mn  mn n  m n

5. DD C( 2nPm) = (12m212 )m n3(20m24m4)n2(8m236m28) .n

6. DD C( 2n1Pm) = 2(2n1)(3m n2 28m n2 3mn25m24mn 7 m n 6).

7. 2 2 2 2 2

( n m) = (1/ 2) ( 1)(8 18 14 13

DD K P n n m n  m n mn  m  mn

30m2n16).

8. DD P K( nm) = (1/ 3)(n1)(m1) (2 mn2mn2n23m n ).

9. DD C( 2nKm) = 2 ((n m23m2)n2(2m23 )m n m m )( 1).

10. 2 2 2 2

2 1

( n m) = (( 3 2) (3 6 2) 2 ) ( 1)(2 1).

DD C K m  m n  m  m n m  m m n

11. DD K C( n m) = (1/ 4)(n1)(m1)(m n2 35m n2 22mn3 14m n2 2mn2

2 2

12m 6mn 4n 8m 4 ).n

    

12. DD P K( n m) = (1/ 3)(n1)(4m n2 2m n2 6mn22n212m n ).

13. DD C( 2nKm) = 4 (2n m n2 23m n2 3mn2m22mn n 22 ).m

14. DD C( 2n1Km) = 2(2n1)(2m n2 25m n2 3mn2 3m25mn n 2n).

15. 2 2 2 2 2

( n m) = (1/ 2) ( 1)(5 11 2 8 2 8 2 2).

DD K K n n m n  m n mn  m  mn m n

Corollary 3.7 We have

1. Gut P P( nm) = (n1)(6m n2 2m26mn30m39).

2. 2 2 2 2 2 2

2

( n m) = 4 (9 12 6 4 4 16 ).

Gut C C n m n  m n mn  m  mn n  m

3. 2 2 2 2 2 2

2 1

( n m) = 2(2 1)(9 21 6 12 10 12 ).

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4. Gut C( 2nPm) = 4 (9n m n2 212m n2 4m26mn20m20). 5. Gut C( 2n1Pm) = 2(2n1)(9m n2 221m n2 12m26mn22m19).

6. 2 2 2 2 2

( n m) = (1/ 3)( 1)(18 12 3 12 2 90 4 3).

Gut P C n m n  mn  m  mn n  m n

7. 2 2 2 2 2

( n m) = (1/ 2) ( 1)(21 46 6 33 16 50 41).

Gut K P n n m n  m n mn  m  mn m n 

8. Gut P K( nm) = (1/ 6)(n1)(m2)(m1) (2 mn2mn2n23m4n3).

9. 2 3 2 2 3 2 2 3 2

( n m) = (1/ 2) (21 67 10 79 30 33

Gut K C n m n  m n  mn  m n mn n  m

3

2mn 22m 3n 1) (3 / 2)n .

    

10. Gut C( 2nKm) = (n m1) (2 m n2 22m n2 4mn24mn4n2m).

11. 2 2 2 2 2

2 1

( n m) = ( 1/ 2)( 1) (( 3 1) (4 8 3) 4 4 ).

Gut C K n m n  n m  n  n m n  n

12. Gut C( 2n1Km) = 2(2n1)(4m n2 28m n2 4mn24m26mn n 2n).

13. Gut C( 2nKm) = 4 (4n m n2 24m n2 4mn2m22mn n 22 ).m

14. Gut P K( nm) = (1/ 3)(n1)(8m n2 24m n2 8mn23m210mn2n2

6m4n3).

15. Gut K K( n m) = (1/ 2) (n n1)(8m n2 217m n2 6mn211m212mn n 2

2m2n1).

A

CKNOWLEDGEMENT

The authors are thankful to the referees for their helpful comments and suggestions. The first author is thankful to the University Grants Commission, Government of India, for the financial support under the Grant F.510/2/SAP-DRS/2011.

R

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Figure

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References

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