Scientific Research PublishingOpen Access Library Journal 2017, Volume 4, e2987 ISSN Online: 2333-9721 ISSN Print: 2333-9705 . Sharp Upper Bounds for Multiplicative Degree Distance

Open Access Library Journal 2017, Volume 4, e2987 ISSN Online: 2333-9721

ISSN Print: 2333-9705

Sharp Upper Bounds for Multiplicative Degree Distance of Graph Operations

R. Muruganandam1, R. S. Manikandan2, M. Aruvi3

1Department of Mathematics, Government Arts College,Tiruchirappalli, India 2Department of Mathematics, Bharathidasan University Constituent College, Lalgudi, Tiruchirappalli, India 3Department of Mathematics, Anna University, Tiruchirappalli, India

Abstract In this paper, multiplicative version of degree distance of a graph is defined and tight upper bounds of the graph operations have been found.

Subject Areas Discrete Mathematics

Keywords Join, Disjunction, Composition, Symmetric Difference, Multiplicative Degree Distance, Zagreb Indices and Coindices

1. Introduction

A topological index of a graph is a numerical quantity which is structural invariant, i.e. it is fixed under graph automorphism. The simplest topological indices are the number of vertices and edges of a graph. In this paper, we define and study a new topological index called multiplicative degree distance. All graphs considered are simple and connected graphs.

We denote the vertex and the edge set of a graph G by ( )V G and ( )E G , respectively. ( )Gd v denotes the degree of a vertex v in G. The number of elements in the vertex set of a graph G is called the order of G and is denoted by ( )v G . The number of elements in the edge set of a graph G is called the size of

G and is denoted by ( )e G . A graph with order n and size m edges is called a ( ),n m -graph. For any ( ),u v V G∈ , the distance between u and v in G, denoted by ( ),Gd u v , is the length of a shortest ( ),u v -path in G. The edge connective the vertices u and v will be denoted by uv. The complement G of the graph G is the graph with vertex set ( )V G , in which two vertices in G are adjacent if

How to cite this paper: Muruganandam, R., Manikandan, R.S. and Aruvi, M. (2017) Sharp Upper Bounds for Multiplicative De- gree Distance of Graph Operations. Open Access Library Journal, 4: e2987. https://doi.org/10.4236/oalib.1102987 Received: August 18, 2016 Accepted: July 8, 2017 Published: July 11, 2017 Copyright © 2017 by authors and Open Access Library Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/

Open Access

DOI: 10.4236/oalib.1102987 July 11, 2017

https://doi.org/10.4236/oalib.1102987http://www.oalib.com/journalhttps://doi.org/10.4236/oalib.1102987http://creativecommons.org/licenses/by/4.0/

R. Muruganandam et al.

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and only if they are not adjacent in G. The join of graphs 1G and 2G is denoted by 1 2G G+ , and it is the graph

with vertex set ( ) ( )1 2V G V G∪ and the edge set ( ) ( ) ( )1 2 1 2E G G E G E G+ = ∪ ( ) ( ){ }1 2 1 1 2 2| , .u u u V G u V G∪ ∈ ∈ The composition of graphs 1G and 2G is

denoted by [ ]1 2G G , and it is the graph with vertex set ( ) ( )1 2V G V G× , and two vertices ( )1 2,u u u= and ( )1 2,v v v= are adjacent if ( 1u is adjacent to 1v ) or ( 1 1u v= and 2u and 2v are adjacent). The disjunction of graphs 1G and 2G is denoted by 1 2G G∨ , and it is the graph with vertex set ( ) ( )1 2V G V G× and ( ) ( )( ) ( ) ( ){ }1 2 1 2 1 2 1 1 1 2 2 2, , | or .E G G u u v v u v E G u v E G∨ = ∈ ∈ The symmetric di-

fference of graphs 1G and 2G is denoted by 1 2G G⊕ , and it is the graph with vertex set ( ) ( )1 2V G V G× and edge set ( ) ( )( ) ( ) ( ){ }1 2 1 2 1 2 1 1 1 2 2 2, , ) | or but not both .E G G u u v v u v E G u v E G⊕ = ∈ ∈ Let G be a connected graph. The Wiener index ( )W G of a graph G is

defined as

( ){ } ( )

( )( )

( ), ,

1, , .2G Gu v V G u v V G

W G d u v d u v⊆ ∈

= =∑ ∑

Dobrynin and Kochetova [1] and Gutman [2] independently proposed a vertex-degree-Weighted version of Wiener index called degree distance or Schultz molecular topological index, which is defined for a connected graph G as

( ){ } ( )

( ) ( ) ( )( )

( ) ( ) ( ), ,

1, , .2G G G G G Gu v V G u v V G

DD G d u v d u d v d u v d u d v⊆ ∈

= + = + ∑ ∑

The Zagreb indices have been introduced more than thirth years ago by Gutman and Trianjestic [3]. The first Zagreb index ( )1M G of a graph G is defined as

( )( )

( ) ( )( )

( )21 .G G Guv E G v V G

M G d u d v d v∈ ∈

= + = ∑ ∑

The second Zagreb index ( )2M G of a graph G is defined as ( )

( )( ) ( )2 .G G

uv E GM G d u d v

∈

= ∑

The Zagreb indices are found to have applications in QSPR and QSAR studies as well, see [4].

Note that contribution of nonadjacent vertex pair should be taken into account when computing the Weighted Wiener Polynomials of certain Composite graphs, see [5]. A.R. Ashrafi, T. Doslic, A. Hamzeha, [6] [7] defined the first Zagreb coindex of a graph G is

( )( )

( ) ( )1 G Guv E G

M G d u d v∉

= + ∑

The second Zagreb coindex of a graph G is

( )( )

( ) ( )2 ,G Guv E G

M G d u d v∉

= ∑

respectively. In [8], Hamzeh, Iranmanesh Hossein-Zadeh and M.V. Diudea recently

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introduced the generalized degree distance of graphs. Asma Hamzeh, Ali Iranmanesh and Samaneh Hossein-Zadeh, Cartesian product, composition, join, disjunction and symmetric difference of graphs and introduce generalized and modified generalized degree distance Polynomials of graphs, such that their first derivatives at 1x = , see [9].

In this paper, we defne a new graph invariant named multiplicative version of degree distance of a graph denoted by ( )*DD G and defined by

( )( )

( ) ( ) ( )2*

, ,, .G G G

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

∈ ≠

= + ∏

Therefore the study of this new topological index is important and we have obtained Sharp upper bounds for the graph operations such as join, disjunction, composition, symmetric difference of graphs.

2. The Multiplicative Degree Distance of Graph Operations

Lemma 2.1. [10] [11], Let 1G and 2G be two simple connected graphs. The number of vertices and edges of graph iG is denoted by in and ie respectively for 1,2i = . Then we have

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

1 2 1 21, or or and,2, otherwise

G G

uv E G uv E G u V G v V Gd u v+

∈ ∈ ∈ ∈=

For a vertex u of 1G , ( ) ( )1 2 1 2 ,G G Gd u d u n+ = + and for a vertex v of 2G , ( ) ( )

1 2 2 1.G G Gd v d v n+ = +

2. [ ] ( ) ( )( )( )

( )1

1 2

1 2 1 2

1 1 2 2 1 2 1 2 2

, ,

, , , 1, ,2, otherwise

G

G G

d u u u u

d u v u v u u v v E G

≠

= = ∈

[ ] ( ) ( ) ( )1 21 2 2, .G GG Gd u v n d u d v= +

3. ( ) ( )( ) ( ) ( )1 2 1 2 1 1 2 21 1 2 21, or

, , ,2, otherwiseG G

u u E G v v E Gd u v u v∨

∈ ∈=

( )( ) ( ) ( ) ( ) ( )1 2 1 2 1 22 1, .G G G G G Gd u v n d u n d v d u d v∨ = + −

4. ( ) ( )( ) ( ) ( )1 2 1 2 1 1 2 21 1 2 21, or but not both

, , ,2, otherwiseG G

u u E G v v E Gd u v u v⊕

∈ ∈=

( )( ) ( ) ( ) ( ) ( )1 2 1 2 1 22 1, 2 .G G G G G Gd u v n d u n d v d u d v⊕ = + − Lemma 2.2. (Arithmetic Geometric inequality) Let 1 2, , , nx x x be non-negative numbers. Then

1 21 2

n nn

x x x x x xn

+ + +≥

Remark 2.3. For a graph G, let ( )A G = { ( ) ( ) ( ), |x y V G V G x∈ × and y are adjacent in G } and let ( )B G = { ( ) ( ) ( ), |x y V G V G x∈ × and y are not adjacent in G }. For each ( ) ( ) ( ), ,x V G x x B G∈ ∈ . Clearly,

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( ) ( ) ( ) ( ).A G B G V G V G∪ = × Let ( ) ( ) ( ){ }, |C G x x x V G= ∈ and ( ) ( ) ( ).D G B G C G= − Clearly ( ) ( ) ( )B G C G D G= ∪ , ( ) ( )C G D G∩ =∅ .

The summation ( ) ( ),x y A G∈∑ runs over the ordered pairs of ( )A G . For simplicity,

we write the summation ( ) ( ),x y A G∈∑ as

xy G∈∑ . Similarly, we write the summation

( ) ( ),x y B G∈∑ as

xy G∉∑ . Also the summation

( )xy E G∈∑ runs over the edges of G. We

denote the summation ( ),x y V G∈

∑ by ,x y G∈∑ and similarly

( ),x y V G∈∑ by

,x y G∈∏ . The

summation ( ) ( ),x y D G∈∑ eqivalent to

,xy G x y∉ ≠∑ and similarly the summation

( ) ( ),x y C G∈∑ eqivalent to

,xy G x y∉ =∑ .

Lemma 2.4. Let G be a graph. Then

( )1 2xy G

e G∈

=∑

Proof:

( )( )1 2 1 2

xy G xy E Ge G

∈ ∈

= =∑ ∑

Lemma 2.5.

( ) ( )1Gxy G

d x M G∈

=∑

Proof: Let ( )x V G∈ and ( )Gt d x= . Let 1 2, , , ty y y be the neighbours of x. Each orderd pair ( ), ,1 ,ix y i t≤ ≤ contributes ( )Gd x to the sum. Thus these orderd pairs contribute ( )2Gd x to the sum. Hence

( )( )

( ) ( )2 1G Gxy G x V G

d x d x M G∈ ∈

= =∑ ∑

Lemma 2.6.

( ) ( ) ( )22G Gxy G

d x d y M G∈

=∑

Proof: Clearly,

( ) ( )( )

( ) ( ) ( )22 2 .G G G Gxy G xy E G

d x d y d x d y M G∈ ∈

= =∑ ∑

Lemma 2.7.

( ) ( )1 2xy G

e G v G∉

= +∑

Proof:

( ) ( ) ( ) ( )( ) ( )

, ,1 1 1 2

xy G x y D G x x C Ge G v G

∉ ∈ ∈

= + = +∑ ∑ ∑

Lemma 2.8.

( ) ( ) ( )( ) ( ) ( )12 1 2Gxy G

d x e G v G e G M G∉

= − + −∑

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Proof.

( )( ) ( )

( )( ) ( )

( )

( ) ( )( ) ( ){ }

( ) ( )( )

( )( )( ) ( ) ( ) ( )

( ) ( )

( )( ) ( )( ) ( )

( ) ( )

( )( ) ( ) ( ) ( )

( ) ( )( ) ( ) ( )

, ,

, ,

, ,

2

,

2

1

1

1 1 2

1 2 2

1 2 2

2 1 2 by Lemma 2.5

G G Gxy G x y D G x x C G

GGx y D G x x C G

Gx y D G x y D G

Gx y A G

Gxy G

d x d x d x

v G d x d x

v G d x e G

v G e G d x e G

v G e G d x e G

e G v G e G M G

∉ ∈ ∈

∈ ∈

∈ ∈

∈

∈

= +

= − − +

= − − +

= − − +

= − − +

= − + −

∑ ∑ ∑

∑ ∑

∑ ∑

∑

∑

Lemma 2.9.

( ) ( ) ( ) ( )2 12G Gxy G

d x d y M G M G∉

= +∑

Proof:

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )

( )( ) ( )

( )( )

( ) ( )

, ,

2

2 1

2

2

G G G G G Gxy G x y D G x x C G

G G Gxy E G x V G

d x d y d x d y d x d x

d x d y d x

M G M G

∉ ∈ ∈

∉ ∈

= +

= +

= +

∑ ∑ ∑

∑ ∑

Lemma 2.10.

( ) ( ) ( ) ( )12 4G Gxy G

d x d y M G e G∉

+ = + ∑

Proof:

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )

( )( )

( )( ) ( )

( ) ( )

, ,

1

2 2

4 2

G G G G G Gxy G x y C G x y D G

G G Gx V G xy E G

d x d y d x d y d x d y

d x d x d y

e G M G

∉ ∈ ∈

∈ ∉

+ = + + +

= + +

= +

∑ ∑ ∑

∑ ∑

3. The Multiplicative Degree Distance of Composition of Graph

Theorem 3.1. Let , 1, 2iG i = , be a ( ),i in m -graph. Then

[ ]( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( )( )( )1 2 1 2

2*1 2

1 2 1 2 2 1 1 1 21 2 1 2

22 1 2 1 1 2 1 2 2 1 1 2

1

2 1 2 2

1 4 4 41

8 1 2 8 4

2 2n n n n

DD G G

M G W G n m DD G n M Gn n n n

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

n DD G m n−

≤ + + −

+ − + + +

+ +

Proof:

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[ ]( )

( )[ ] ( ) ( )( ) [ ] ( ) [ ] ( )

( ) ( ) [ ]( ) ( )( ) [ ] ( ) [ ] ( )

( )

( ) [ ] ( ) ( )( ) [ ]

1 2 1 2 1 21 2

1 2 1 2

1 2 1 2 1 21 2

1 2 1 21 2

2*1 2

, , , or1

, , , or1 2 1 2

, ,1 2 1 2

, , , , ,

1 , , , , ,1

1 , , ,1

G G G G G Gx y G u v G x u y v

n n n n

G G G G G Gx y G u v G x u y v

G G G Gx y G u v G

DD G G

d x u y v d x u d y v

d x u y v d x u d y vn n n n

d x u y v dn n n n

∈ ∈ ≠ ≠−

∈ ∈ ≠ ≠

∈ ∈

= +

≤ + −

=−

∏ ∏

∑ ∑

∑ ∑ ( ) [ ] ( )( )

( ) [ ] ( ) ( )( ) [ ] ( ) [ ] ( )( )

[ ] ( ) ( )( ) [ ] ( ) [ ] ( )( )( )

( ) [ ] ( ) ( )( ) [ ]

1 2 1 2

1 2

1 2 1 2 1 21 2

1 2 1 2

1 2 1 2 1 22

1 2 1 21 2

1

,1 2 1 21

, ,1 2 1 2

, ,

1 , , , , ,1

, , , , ,

1 , , , ,1

n n n n

G G

G G G G G Gx y G uv G

n n n n

G G G G G Guv G

G G G Gx y G x y uv G

x u d y v



d x u y v d xn n n n

−

∈ ∈

−

∉

∈ = ∈

+ = + −

+ +

=−

∑ ∑

∑

∑ ∑ ( ) [ ] ( )

[ ] ( ) ( )( ) [ ] ( ) [ ] ( )

[ ] ( ) ( )( ) ( ) [ ] ( )

[ ] ( ) ( )( ) [ ] ( ) [ ] ( )( )

1 2

1 2 1 2 1 21 2

1 21 2 1 21 2

1 2 1 2

1 2 1 2 1 21 2

, ,

[ ], ,

1

, ,

,

, , , , ,

, , , , ,

, , , , ,

G G

G G G G G Gx y G x y uv G

G GG G G Gx y G x y uv G

n n n n


u d y v




∈ ≠ ∈

∈ = ∉−

∈ ≠ ∉

+ + +

+ +

+ +

∑ ∑

∑ ∑

∑ ∑

[ ] ( ) ( )( )1 2 1 2 1

2*1 2 3 1 2 4

1 2 1 2

11

n n n n

DD G G S S S Sn n n n

− ≤ + + + −

where 3 1 2 4, , ,S S S S are terms of the above sums taken in order. Next we calculate 1 2 3, ,S S S and 4S separately.

[ ] ( ) ( )( ) [ ] ( ) [ ] ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

1 2 1 2 1 21 2

1 2 1 2 11 2

1 1 1 1 2 21 2 1 2

1, ,

2 2, ,

2, , , ,

2 2

, , , , ,

, by Lemma 2.1

, ( ) 1 ,

4


G G G G Gx y G x y uv G

G G G G G Gx y G x y uv G x y G x y uv G

S d x u y v d x u d y v

d x y d u n d x d v n d y

n d x y d x d y d x y d u d v

n m D

∈ ≠ ∈

∈ ≠ ∈

∈ ≠ ∈ ∈ ≠ ∈

= +

= + + +

= + + +

=

∑ ∑

∑ ∑

∑ ∑ ∑ ∑( ) ( ) ( )1 1 2 14D G M G W G+

[ ] ( ) ( )( ) [ ] ( ) [ ] ( )

[ ] ( ) ( )( ) [ ] ( ) [ ] ( )

[ ] ( ) ( )( ) [ ] ( ) [ ] ( )

[ ] ( ) ( )( )

1 2 1 2 1 21 2

1 2 1 2 1 21 2

1 2 1 2 1 22

1 21 2

2, ,

, , ,

,

, , ,

, , , , ,

, , , , ,

, , , , ,

0 , , ,


G G G G G Gx y G x y uv G u v

G G G G G Guv G u v

G G Gx y G x y uv G u v




d x u y v d

∈ = ∉

∈ = ∉ =

∉ ≠

∈ = ∉ ≠

= +

= + + +

= +

∑ ∑

∑ ∑

∑

∑ ∑ [ ] ( ) [ ] ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

1 2 1 2

1 2 1 2 11 2

2 2 1 12 1 2 1

2 2, , ,

2, , , , , ,

1 1 2 2 1 2 2

, ,

, by Lemma 2.1

2 1 2 1

4 8 1

G G G

G G G G Gx y G x y uv G u v

G G G Guv G u v x y G x y uv G u v x y G x y

x u d y v

d x y d u n d x d v n d y

d u d v n d x d y

n M G n m n n

∈ = ∉ ≠

∉ ≠ ∈ = ∉ ≠ ∈ =

+

= + + +

= + + +

= + −

∑ ∑

∑ ∑ ∑ ∑

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[ ] ( ) ( )( ) [ ] ( ) [ ] ( )

[ ] ( ) [ ] ( )

( ) ( ) ( ) ( )

( ) ( )

1 2 1 2 1 21 2

1 2 1 21 2

2 1 2 11 2

2 21 2 1

3, ,

, ,

2 2, ,

2, , , ,

, , , , ,

1 , ,

by Lemma 2.1

1




G Gx y G x y uv G x y G x y


d x u d y v

d u d x n d v n d y

d u d v n

∈ = ∈

∈ = ∈

∈ = ∈

∈ = ∈ ∈ =

= +

= ⋅ +

= + + +

= + +

∑ ∑

∑ ∑

∑ ∑

∑ ∑ ( ) ( )

( )

1 12

1 1 2 2 1 2

1

2 8

G Guv G

d x d y

n M G n m m∈

+

= +

∑ ∑

[ ] ( ) ( )( ) [ ] ( ) [ ] ( )

( ) [ ] ( ) [ ] ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

1 2 1 2 1 21 2

1 1 2 1 21 2

1 2 1 2 11 2

1 2 21 2

4, ,

, ,

2 2, ,

, ,

, , , , ,

, , ,

, by Lemma 2.1

,




G G Gx y G x y uv G


d x y d x u d y v

d x y d u d x n d v n d y

d x y d u d v

∈ ≠ ∉

∈ ≠ ∉

∈ ≠ ∉

∈ ≠ ∉

= +

= +

= + + +

= +

∑ ∑

∑ ∑

∑ ∑

∑ ∑

( ) ( ) ( )

( ) ( ) ( )( )1 1 1

1 2

2, ,

1 1 2 2 1 2 2

, 1

4 2 2

G G Gx y G x y uv G

n d x y d x d y

W G M G n DD G m n∈ ≠ ∉

+ +

= + +

∑ ∑

[ ]( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( )( )( )1 2 1 2

2*1 2 1 2 1 2 2 1 1 1 2

1 2 1 2

22 1 2 1 1 2 1 2 2 1 1 2

1

2 1 2 2

1 4 4 41

8 1 2 8 4

2 2n n n n

DD G G M G W G n m DD G n M Gn n n n

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

n DD G m n−

≤ + + −+ − + + +

+ +

Lemma 3.2.

[ ] ( )( )1

* 22 1nr nr

n rDD K K nr−

= −

Proof: Clearly the graph [ ]n rK K is the complete graph nrK .

[ ]( ) ( ) ( )( )1

* * 22 1nr nr

n r nrDD K K DD K nr−

∴ = = − (1)

Remark 3.3. Let 1 nG K= and 2 rG K= . We get,

( ) ( ) ( ) ( ) ( ) ( ) ( )21 1 11 1 1

2 1 1 , ,2 2 2

n n n n n nDD G n n n m W G

− − −= − = − = =

( ) ( ) ( ) ( )21 2 1 2 2 1 2 21

1 , 0, 0, , ,2

r rM G r r M G m n n n r m

−= − = = = = =

∴ In Theorem 3.1, put 1 nG K= and 2 rG K= , we get

[ ] ( )( )1

* 22 1nr nr

n rDD K K nr−

≤ − (2)

From (1) and (2) our bound is tight

4. The Multiplicative Degree Distance of Join of Graph

Theorem 4.1. Let , 1, 2iG i = , be a ( ),i in m -graph and let ( )i im e G= . Then

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( ) ( )( ) ( ) ( )

( ) ( )

( )( )( )1 2 1 2

2*1 2 1 1 2 1 1 1 2 1

1 2 1 2

1 2 2 1 1 2 1 2 1 2

1

1 2 1 2 1 2

1 2 4 4 81

2 2 2

4 4 8n n n n

DD G G M G n m M G n mn n n n

m n m n n n n n M G

n m M G n m+ + −

+ ≤ + + + + + −+ + + + +

+ + +

Proof:

( )

( )( ) ( ) ( ) ( ) ( ) ( )

( )( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( )( )

( )( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( )

1 2 1 2 1 21 2

1 2 1 2

1 2 1 2 1 21 2

1 2

1 2 1 2 1 21 2

2*1 2

, ,

1

, ,1 2 1 2

,1 2 1 2

,

1 ,1

1 ,1

G G G G G Gx y V G G x y

n n n n

G G G G G Gx y V G G x y

n n

G G G G G Gx y V G G

DD G G

d x y d x d y

d x y d x d yn n n n


+ + +∈ + ≠

+ + −

+ + +∈ + ≠

+

+ + +∈ +

+

= +

≤ + + + −

= + + + −

∏

∑

∑( )

( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( )( )

( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( )( ) ( ) ( ) ( ) ( ) ( )

1 2

1 2 1 2

1 2 1 2 1 21 2 1 2

1 2 1 2 1 21 2 1

1

1 2 1 2 1 22

1

1

1 2 1 2

1 2 1 2

1 ,1

1 ,1

,

n n

n n n n

G G G G G Gx V G G y V G G

G G G G G Gx V G G y V G

n

G G G G G Gy V G



d x y d x d y

+ −

+ + −

+ + +∈ + ∈ +

+ + +∈ + ∈

+

+ + +∈

= + + + −

= + + + −

+ +

∑ ∑

∑ ∑

∑( )( )

( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( )( )

( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( )

2 1 2

1 2 1 2 1 21 2 1

1 2 1 2

1 2 1 2 1 21 2 2

1 2 1 2 1 21 1

2 1

1

1 2 1 2

1

1 2 1 2

1 ,1

,

1 ,1

n n n


n n n n


G G G G G Gx V G y V G

x V G y V G


d x y d x d y


+ −

+ + +∈ + ∈

+ + −

+ + +∈ + ∈

+ + +∈ ∈

∈ ∈

= + + + −

+ +

= + + + −

+

∑ ∑

∑ ∑

∑ ∑

∑ ( ) ( ) ( ) ( ) ( ) ( )

( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( )( )

( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( )

1 2 1 2 1 2

1 2 1 2 1 21 2

1 2 1 2

1 2 1 2 1 22 2

1 2 1 2 1 21 1

1 2

1

1 2 1 2

,

,

,

1 ,1

2

G G G G G G


n n n n



x V G y V G

d x y d x d y

d x y d x d y

d x y d x d y


d

+ + +

+ + +∈ ∈

+ + −

+ + +∈ ∈

+ + +∈ ∈

∈ ∈

+

+ +

+ +

= + + + −

+

∑

∑ ∑

∑ ∑

∑ ∑

∑ ∑ ( ) ( ) ( ) ( ) ( ) ( )

( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( )( )

1 2 1 2 1 2

1 2 1 2

1 2 1 2 1 22 2

1

,

,

G G G G G G

n n n n


x y d x d y

d x y d x d y

+ + +

+ + −

+ + +∈ ∈

+

+ +

∑ ∑

8/18


OALib Journal

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

2*1 2 1 2 3

1 2 1 2

1 21

n n n n

DD G G J J Jn n n n

+ + −

+ ≤ + + + + −

where 1 2 3, ,J J J are terms of the above sums taken in order. Next we calculate 1 2,J J and 3J separately one by one. Now,

( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 2 1 2 1 21 1

1 2 1 2 1 21

1 2 1 2 1 21

1 2 1 2 1 21

1 2 1 2 1 21

1

,

,

,

,

,

,

,

,

1


G G G G G Gx y V G

G G G G G Gxy G

G G G G G Gxy G x y

G G G G G Gxy G x y

xy G

J d x y d x d y

d x y d x d y

d x y d x d y

d x y d x d y

d x y d x d y

+ + +∈ ∈

+ + +∈

+ + +∈

+ + +∉ ≠

+ + +∉ =

∈

= +

= +

= +

+ +

+ +

= ⋅

∑ ∑

∑

∑

∑

∑

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

1 2 1 21

1 2 1 21

1 21

1 21

1 21 1

1 21 1

,

2 2

2 2,

2

2, ,

2 0

2 by Lemma 2.1

2 1

2 2 1

G G G G

G G G Gxy G x y

G Gxy G

G Gxy G x y

G Gxy G xy G

G Gxy G x y xy G x y

d x d y

d x d y

d x n d y n

d x n d y n

d x d y n

d x d y n

+ +

+ +∉ ≠

∈

∉ ≠

∈ ∈

∉ ≠ ∉ ≠

+

+ ⋅ + +

= + + +

+ + + +

= + +

+ + +

∑

∑

∑

∑

∑ ∑

∑ ∑

( ) ( )1 1 2 1 1 1 2 12 4 4 8M G n m M G n m

= + + +

( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( )( ) ( ) ( ) ( )

( ) ( )( ) ( )

( ) ( ) ( ) ( )( ) ( )

( ) ( )

( )

1 2 1 2 1 21 2

1 2 1 21 2

1 21 2

1 21 2 1 2 1 2

2

2 1

1 2

1 2 2 1 1 2 1 2

,

1

by Lemma 2.1

( ) 1 1 1 1

2 2


G G G Gx V G y V G

G Gx V G y V G

G Gx V G y V G x V G y V G x V G y V G

J d x y d x d y

d x d y

d x n d y n

d x d y n n

m n m n n n n n

+ + +∈ ∈

+ +∈ ∈

∈ ∈

∈ ∈ ∈ ∈ ∈ ∈

= +

= +

= + + +

= + + +

= + + +

∑ ∑

∑ ∑

∑ ∑

∑ ∑ ∑ ∑ ∑ ∑

( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 2 1 2 1 22 2

1 2 1 2 1 22

1 2 1 2 1 22

1 2 1 2 1 22

1 2 1 2 1 22

3

,

,

,

,

,

,

,

,


G G G G G Gx y V G

G G G G G Gxy G

G G G G G Gxy G x y

G G G G G Gxy G x y

J d x y d x d y

d x y d x d y

d x y d x d y

d x y d x d y

d x y d x d y

+ + +∈ ∈

+ + +∈

+ + +∈

+ + +∉ ≠

+ + +∉ =

= +

= +

= +

+ +

+ +

∑ ∑

∑

∑

∑

∑

9/18


OALib Journal

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

1 2 1 22

1 2 1 22

2 22

2 22

2 22 2

2 22 2

,

1 1

1 1,

1

1, ,

1

2 0

2 by Lemma 2.1

2 1

2 2 1

G G G Gxy G

G G G Gxy G x y

G Gxy G

G Gxy G x y

G Gxy G xy G

G Gxy G x y xy G x y

d x d y

d x d y

d x n d y n

d x n d y n

d x d y n

d x d y n

+ +∈

+ +∉ ≠

∈

∉ ≠

∈ ∈

∉ ≠ ∉ ≠

= +

+ + +

= + + +

+ + + +

= + +

+ + +

∑

∑

∑

∑

∑ ∑

∑ ∑

( ) ( )1 2 1 2 1 2 1 22 4 4 8M G n m M G n m

= + + +

( ) ( )( ) ( ) ( )

( ) ( )

( )( )( )1 2 1 2

2*1 2 1 1 2 1 1 1 2 1

1 2 1 2

1 2 2 1 1 2 1 2 1 2

1

1 2 1 2 1 2

1 2 4 4 81

2 2 2

4 4 8n n n n

DD G G M G n m M G n mn n n n

m n m n n n n n M G

n m M G n m+ + −

+ ≤ + + + + + −+ + + + +

+ + +

Lemma 4.2.

[ ]

( )( )( )

*

122 1

n r

n r n r

DD K K

n r+ + −

+

= + −

Proof: Clearly the graph n rK K+ is the complete graph n rK +

[ ][ ]

( )( )( )

*

*

12

=

2 1

n r

n r

n r n r

DD K K

DD K

n r

+

+ + −

+

= + −

(3)

Remark 4.3. Let 1 nG K= and 2 rG K= . We get,

( ) ( )( )

( ) ( )( )

21 1

1

21 2

1 2

1 ,

1,

21 ,

0,

M G n n

n nm

M G r r

M G

= −

−=

= −

=

( ) ( )2 1 1 1 2 1 21

, 0, , , 0, 0.2

r rm M G n n n r m m

−= = = = = =

∴ In Theorem 4.1, put ,=,= 21 rn KGKG we get

[ ] ( )( )( )1

* 22 1n r n r

n rDD K K n r+ + −

+ ≤ + − (4)

From (3) and (4) our bound is tight.

5. The Multiplicative Degree Distance of Disjunction of Graph

Theorem 5.1. Let , 1, 2iG i = , be a ( ),i in m -graph and let ( )i im e G= . Then

10/18


OALib Journal

( )

( ) ( )( ) ( ) ( ){

( ) ( ) ( )( ) ( )( )( )( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )

( )( )( ) ( )( )( )

2*1 2

2 2 1 1 1 1 1 1 1 21 2 1 2

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

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

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

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

1 2 2 4 2 21

2 2 1 2 2 2

2 2 4 2 2 1 2

4 4 2

2 2 4 2 2 2 2 4 2

DD G G

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

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

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

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

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

∨

≤ + + +

−

− − + − + +

+ + − − + −

+ + −

+ + + + + +

− ( ) ( )( ) ( ) ( )( )

}( )1 2 1 2

1 1 1 1 1 2 2 2 1 2

12 2

1 2 1 2 1 2

4 2 1 2 2 1 2

8 4 16n n n n

m n m M G m n m M G

m n n m m m−

− + − − + −

− − +

Proof:

( )

( )( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( )

( )

( )

1 2 1 2 1 21 2

1 2 1 2

1 2 1 2 1 21 2

1 2

2*1 2

, , , or

1

, , , or1 2 1 2

, ,1 2 1 2

, , , , ,

1 , , , , ,1

11

G G G G G Gx y G u v G x y u v

n n n n


Gx y G u v G

DD G G



dn n n n

∨ ∨ ∨∈ ∈ ≠ ≠

−

∨ ∨ ∨∈ ∈ ≠ ≠

∈ ∈

∨

= +

≤ + −

=−

∏ ∏

∑ ∑

∑ ∑ ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( ) ( )( )

1 2 1 2

1 2 1 2 1 2

1 2 1 2 1 21 2

1 2 1 2

1 2 1 2 1 22

1

,1 2 1 2

1

, , , , ,

1 , , , , ,1

, , , , ,

1

n n n n

G G G G G


n n n n

G G G G G Guv G

x u y v d x u d y v



−

∨ ∨ ∨

∨ ∨ ∨∈ ∈

−

∨ ∨ ∨∉

+

= + −

+ +

=

∑ ∑

∑

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )

1 2 1 2 1 21 2

1 2 1 2 1 21 2

1 2 1 2 1 21 2

1 2 1 21 2

1 2 1 2

, , , , ,1

, , , , ,

, , , , ,

, , , ,

G G G G G Gxy G uv G



G G G Gxy G uv G




d x u y v d x u

∨ ∨ ∨∈ ∈

∨ ∨ ∨∉ ∈

∨ ∨ ∨∈ ∉

∨ ∨∉ ∉

+ −

+ +

+ +

+ +

∑ ∑

∑ ∑

∑ ∑

∑ ∑ ( ) ( )( )1 2 1 2

1 2

1

,n n n n

G Gd y v−

∨

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

2*1 2 3 1 2 4

1 2 1 2

11

n n n n

DD G G A A A An n n n

−

∨ ≤ + + + −

where 3 1 2 4, , ,A A A A are terms of the above sums taken in order. Next we calculate 1 2 3, ,A A A and 4A separately one by one. Now,

11/18


OALib Journal

( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )

( )

1 2 1 2 1 21 2

1 2 1 2 11 2

2 1 2

1 1 2 21 2 1 2

11 2

1

2 1 2

1

2 1

, , , , ,

1

by Lemma 2.1


G G G G Gxy G uv G

G G G

G G G Gxy G uv G xy G uv G

G Gxy G uv G

A d x u y v d x u d y v

n d x n d u d x d u n d y

n d v d y d v

n d x d y n d u d v

d x d

∨ ∨ ∨∉ ∈

∉ ∈

∉ ∈ ∉ ∈

∉ ∈

= +

= ⋅ + − +

+ − = + + +

−

∑ ∑

∑ ∑

∑ ∑ ∑ ∑

∑ ∑ ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( )( ) ( ) ( )

2 1 21 2

1 1 2 21 2 1 2

1 2 1 21 2 1 2

2 1

2 2 1 1 1 1 1 1 1 2

1 1

2 2 4 2 2

G Gxy G uv G



u d y d v

n d x d y n d u d v

d x d u d y d v

m n M G m n m n M G

∉ ∈

∉ ∈ ∉ ∈

∉ ∈ ∉ ∈

−

= + + +

− −

= + + +

∑ ∑

∑ ∑ ∑ ∑

∑ ∑ ∑ ∑

( ) ( ) ( )( )1 2 1 1 1 1 12 2 1 2M G m n m M G− − + −

( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( )

1 2 1 2 1 21 2

1 2 1 2 11 2

2 1 2

1 2 2 21 2 1 2

1 21 2

2

2 1 2

1

2 1

, , , , ,

by Lemma 2.1


G G G G Gxy G uv G

G G G


G Gxy G uv G



n d v d y d v

n d x d y n d u d v

d x d u

∨ ∨ ∨∈ ∉

∈ ∉

∈ ∉ ∈ ∉

∈ ∉

= +

= + − +

+ − = + + +

−

∑ ∑

∑ ∑

∑ ∑ ∑ ∑

∑ ∑ ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( )( ) ( )( )

1 21 2

1 2 2 21 2 1 2

1 2 1 21 2 1 2

2 1

2 1 1 2 2 1 1 1 2 2

1 1

2 2 2 2 4

2

G Gxy G uv G



d y d v

n d x d y n d u d v

d x d u d y d v

n M G m n n m M G m

∈ ∉

∈ ∉ ∈ ∉

∈ ∉ ∈ ∉

−

= + + +

− −

= + + +

−

∑ ∑

∑ ∑ ∑ ∑

∑ ∑ ∑ ∑

( ) ( ) ( )1 1 2 2 2 1 22 1 2M G n m m M G − + −

( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( )

1 2 1 2 1 21 2

1 2 1 2 11 2

2 1 2

1 2 2 21 2 1 2

1 21 2

3

2 1 2

1

2 1

, , , , ,

by Lemma 2.1


G G G G Gxy G uv G

G G G


G Gxy G uv G



n d v d y d v

n d x d y n d u d v

d x d u

∨ ∨ ∨∈ ∈

∈ ∈

∈ ∈ ∈ ∈

∈ ∈

= +

= + − +

+ − = + + +

−

∑ ∑

∑ ∑

∑ ∑ ∑ ∑

∑ ∑ ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

1 21 2

1 2 2 21 2 1 2

1 2 1 21 2 1 2

2 1

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

1 1

4 4 2

G Gxy G uv G



d y d v

n d x d y n d u d v

d x d u d y d v

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

∈ ∈

∈ ∈ ∈ ∈

∈ ∈ ∈ ∈

−

= + + +

− −

= + −

∑ ∑

∑ ∑ ∑ ∑

∑ ∑ ∑ ∑

12/18


OALib Journal

( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

1 2 1 2 1 21 2

1 2 1 21 2

1 2 1 21 2

4

, ,

, , , , ,

2 , ,

2 , ,


G G G Gxy G uv G

G G G Gxy G x y uv G u v


d x u d y v

d x u d y v

∨ ∨ ∨∉ ∉

∨ ∨∉ ∉

∨ ∨∉ = ∉ =

= +

= +

− +

∑ ∑

∑ ∑

∑ ∑

4 5 62 2 ,A A A= − where 5A and 6A are terms of the above sums taken in order.

Now,

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

1 2 1 21 2

1 2 1 2 11 2

2 1 2

1 1 2 21 2 1 2

1 21 2 1 2

5

2 1 2

1

2 1

, ,

by Lemma 2.1

G G G Gxy G uv G

G G G G Gxy G uv G

G G G


G Gxy G uv G xy G uv G

A d x u d y v


n d v d y d v

n d x d y n d u d v

d x d u

∨ ∨∉ ∉

∉ ∉

∉ ∉ ∉ ∉

∉ ∉ ∉ ∉

= +

= + − +

+ −

= + + +

− −

∑ ∑

∑ ∑

∑ ∑ ∑ ∑

∑ ∑ ∑ ∑ ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( )( )( ) ( )( )( )

1 2

1 2 2 21 2 1 2

1 2 1 21 2 1 2

2 1

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

1 1

1 1

2 4 2 2 4 2

2 2

G G



d y d v

n d x d y n d u d v

d x d u d y d v


m n

∉ ∉ ∉ ∉

∉ ∉ ∉ ∉

= + + +

− −

= + + + + +

− −

∑ ∑ ∑ ∑

∑ ∑ ∑ ∑

( ) ( )( ) ( ) ( )( )1 1 1 2 2 2 1 21 2 2 1 2m M G m n m M G+ − − + −

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

1 2 1 21 2

1 2 1 2 11 2

2 1 2

1 1 2 21 2 1 2

6, ,

2 1 2, ,

1

2 1, , , ,

, ,

by Lemma 2.1


G G G G Gxy G u v uv G u v

G G G

G G G Gxy G x y uv G u v xy G x y uv G u v

x

A d x u d y v


n d v d y d v

n d x d y n d u d v

∨ ∨∉ = ∉ =

∉ = ∉ =

∉ = ∉ = ∉ = ∉ =

= +

= + − +

+ −

= + + +

−

∑ ∑

∑ ∑

∑ ∑ ∑ ∑

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

1 2 1 21 2 1 2

1 2 2 21 2 1 2

1 21 2

, , , ,

2 1, , , ,

, ,

1 1

G G G Gy G x y uv G u v xy G x y uv G u v


G Gxy G x y uv G u v xy G

d x d u d y d v

n d x d y n d u d v

d x d u

∉ = ∉ = ∉ = ∉ =

∉ = ∉ = ∉ = ∉ =

∉ = ∉ = ∉

−

= + + +

− −

∑ ∑ ∑ ∑

∑ ∑ ∑ ∑

∑ ∑ ( ) ( )1 21 2, ,

2 21 2 2 1 1 24 4 8

G Gx y uv G u v

d y d v

m n m n m m

= ∉ =

= + −

∑ ∑

13/18


OALib Journal

( )

( ) ( )( ) ( ) ( ){( ) ( ) ( )( ) ( )( )

( )( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )

( )( )( ) ( )( )( )

2*1 2

2 2 1 1 1 1 1 1 1 21 2 1 2

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

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

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

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

1 2 2 4 2 21

2 2 1 2 2 2

2 2 4 2 2 1 24 4 22 2 4 2 2 2 2 4 2

DD G G

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


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

∨

≤ + + +−

− − + − + +

+ + − − + −

+ + −+ + + + + +

− ( ) ( )( ) ( ) ( )( )}

( )1 2 1 21 1 1 1 1 2 2 2 1 2

12 2

1 2 1 2 1 2

4 2 1 2 2 1 2

8 4 16n n n n

m n m M G m n m M G

m n n m m m−

− + − − + −

− − +

Lemma 5.2.

[ ] ( )( )1

* 22 2mn mn

m nDD K K mn−

∨ = −

Proof: Clearly the graph m nK K∨ is the complete graph mnK .

( ) ( ) ( )( )1

* * 22 2mn mn

m n mnDD K K DD K mn−

∨ = = − (5)

Remark 5.3. Let 1 mG K= and 2 nG K= . We get

( ) ( )1 2 1 2 1 2

1 1, , , , 0, 0

2 2m m n n

n m n n m m m m− −

= = = = = =

( ) ( ) ( ) ( ) ( ) ( )2 21 1 1 1 2 11 , 1m nM G M K m m M G M K n n= = − = = −

( ) ( ) ( ) ( ) ( ) ( )1 1 1 1 2 1 1 1 10, 0, 0m n mM G M K M G M K M G M K= = = = = = ∴ In Theorem 5.1, put 1 mG K= and 2 nG K= , we get

[ ] ( )( )1

* 22 2mn mn

m nDD K K mn−

∨ ≤ − (6)


6. The Multiplicative Degree Distance of Symmetric difference of Graph

Theorem 6.1.

( )

( ) ( )( ) ( )( )

( ) ( ) ( )( ) ( )( )( )( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )( )( ) ( )( )( )

2*1 2

2 2 1 1 1 1 1 2 1 11 2 1 2

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

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

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

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

1 2 2 4 2 21

4 2 1 2 2 2

2 2 4 4 2 1 24 4 42 2 4 2 2 4 2

DD G G

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


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


⊕ ≤ + + + −− − + − + +

+ + − − + −

+ + −

+ + + + + +

( ) ( )( ) ( ) ( )( )( )

( )1 2 1 2

1 1 1 1 1 2 2 2 1 2

12 22 1 1 2 1 2

4 2 1 2 2 1 2

2 4 4 16n n n n

m n m M G m n m M G

n m n m m m−

− − + − − + −

− + −

14/18


OALib Journal

Proof:

( )

( )( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( )

( )

( )

1 2 1 2 1 21 2

1 2 1 2

1 2 1 2 1 21 2

1 2

2*1 2

, , , or

1

, , , or1 2 1 2

, ,1 2 1 2

, , , , ,

1 , , , , ,1

11


n n n n


Gx y G u v G

DD G G



dn n n n

⊕ ⊕ ⊕∈ ∈ ≠ ≠

−

⊕ ⊕ ⊕∈ ∈ ≠ ≠

∈ ∈

⊕ = +

≤ + −

=−

∏ ∏

∑ ∑

∑ ∑ ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( ) ( )( )

1 2 1 2

1 2 1 2 1 2

1 2 1 2 1 21 2

1 2 1 2

1 2 1 2 1 22

1

,1 2 1 2

1

, , , , ,

1 , , , , ,1

, , , , ,

1

n n n n

G G G G G


n n n n

G G G G G Guv G

x u y v d x u d y v



−

⊕ ⊕ ⊕

⊕ ⊕ ⊕∈ ∈

−

⊕ ⊕ ⊕∉

+ = + −

+ +

=

∑ ∑

∑

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )

1 2 1 2 1 21 2

1 2 1 2 1 21 2

1 2 1 2 1 21 2

1 2 1 21 2

1 2 1 2

, , , , ,1

, , , , ,

, , , , ,

, , , ,




G G G Gxy G uv G




d x u y v d x u

⊕ ⊕ ⊕∈ ∈

⊕ ⊕ ⊕∉ ∈

⊕ ⊕ ⊕∈ ∉

⊕ ⊕∉ ∉

+ − + +

+ +

+ +

∑ ∑

∑ ∑

∑ ∑

∑ ∑ ( ) ( )( )1 2 1 2

1 2

1

,n n n n

G Gd y v−

⊕

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

2*1 2 3 1 2 4

1 2 1 2

11

n n n n

DD G G C C C Cn n n n

−

⊕ ≤ + + + −

where 3 1 2 4, , ,C C C C are terms of the above sums taken in order. Next we calculate 1 2 3, ,C C C and 4C separately.

( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

1 2 1 2 1 21 2

1 2 1 2 11 2

2 1 2

1 1 2 21 2 1 2

11 2

1

2 1 2

1

2 1

, , , , ,

1 2

2 by Lemma 2.1

2


G G G G Gxy G uv G

G G G


Gxy G uv G

C d x u y v d x u d y v


n d v d y d v

n d x d y n d u d v

d

⊕ ⊕ ⊕∉ ∈

∉ ∈

∉ ∈ ∉ ∈

∉ ∈

= +

= ⋅ + − +

+ −

= + + +

−

∑ ∑

∑ ∑

∑ ∑ ∑ ∑

∑ ∑ ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( )( ) ( )

2 1 21 2

1 1 2 21 2 1 2

1 2 1 21 2 1 2

2 1

2 2 1 1 1 1 1 2

2

1 1

2 2

2 2 4 2

G G Gxy G uv G



x d u d y d v

n d x d y n d u d v

d x d u d y d v

n m M G m n M G

∉ ∈

∉ ∈ ∉ ∈

∉ ∈ ∉ ∈

−

= + + +

− −

= + +

∑ ∑

∑ ∑ ∑ ∑

∑ ∑ ∑ ∑

( )

( ) ( ) ( )( )1 1

1 2 1 1 1 1 1

2

4 2 1 2

m n

M G m n m M G

+

− − + −

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OALib Journal

( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( )

1 2 1 2 1 21 2

1 2 1 21 2

1 2 1 2

1 2 2 21 2 1 2

11 2

2

2 1

2 1

2 1

, , , , ,

2

2 by Lemma 2.1

2


G G G Gxy G uv G

G G G G


Gxy G uv G


n d x n d u d x d u

n d y n d v d y d v

n d x d y n d u d v

d x d

⊕ ⊕ ⊕∈ ∉

∈ ∉

∈ ∉ ∈ ∉

∈ ∉

= +

= + −

+ + −

= + + +

−

∑ ∑

∑ ∑

∑ ∑ ∑ ∑

∑ ∑ ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( )( ) ( )

2 1 21 2

1 2 2 21 2 1 2

1 2 1 21 2 1 2

2 1

2 1 1 2 2 1 1 1 2

2

1 1

2 2

2 2 2

G G Gxy G uv G



u d y d v

n d x d y n d u d v

d x d u d y d v

n M G m n n m M G

∈ ∉

∈ ∉ ∈ ∉

∈ ∉ ∈ ∉

−

= + + +

− −

= + + +

∑ ∑

∑ ∑ ∑ ∑

∑ ∑ ∑ ∑

( )

( ) ( ) ( )( )2

1 1 2 2 2 1 2

4

4 2 1 2

m

M G n m m M G− − + −

( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( )

1 2 1 2 1 21 2

1 2 1 21 2

1 2 1 2

1 2 2 21 2 1 2

11 2

3

2 1

2 1

2 1

, , , , ,

2

2 by Lemma 2.1

2


G G G Gxy G uv G

G G G G


Gxy G uv G


n d x n d u d x d u

n d y n d v d y d v

n d x d y n d u d v

d x d

⊕ ⊕ ⊕∈ ∈

∈ ∈

∈ ∈ ∈ ∈

∈ ∈

= +

= + −

+ + −

= + + +

−

∑ ∑

∑ ∑

∑ ∑ ∑ ∑

∑ ∑ ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

2 1 21 2

1 2 2 21 2 1 2

1 2 1 21 2 1 2

2 1

2 1 1 2 1 1 1 2 1

2

1 1

2 2

4 4 4

G G Gxy G uv G



u d y d v

n d x d y n d u d v

d x d u d y d v

n M G m n m M G M G

∈ ∈

∈ ∈ ∈ ∈

∈ ∈ ∈ ∈

−

= + + +

− −

= + −

∑ ∑

∑ ∑ ∑ ∑

∑ ∑ ∑ ∑

( ) ( )1 1 2M G

( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

1 2 1 2 1 21 2

1 2 1 21 2

1 2 1 21 2

4

, ,

, , , , ,

2 , ,

2 , ,


G G G Gxy G uv G



d x u d y v

d x u d y v

⊕ ⊕ ⊕∉ ∉

⊕ ⊕∉ ∉

⊕ ⊕∉ = ∉ =

= +

= +

− +

∑ ∑

∑ ∑

∑ ∑

4 5 62 2 ,C C C= − where 5C and 6C denote the sums of the above terms in order.

Now,

16/18


OALib Journal

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

1 2 1 21 2

1 2 1 21 2

1 2 1 2

1 2 2 21 2 1 2

1 21 2

5

2 1

2 1

2 1

, ,

2

2 by Lemma 2.1

2 ( ) ( ) 2

G G G Gxy G uv G

G G G Gxy G uv G

G G G G


G Gxy G uv G xy G

C d x u d y v

n d x n d u d x d u

n d y n d v d y d v

n d x d y n d u d v

d x d u

⊕ ⊕∉ ∉

∉ ∉

∉ ∉ ∉ ∉

∉ ∉ ∉

= +

= + −

+ + − = + + +

− −

∑ ∑

∑ ∑

∑ ∑ ∑ ∑

∑ ∑

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( )( )( ) ( )( )

1 21 2

1 2 2 21 2 1 2

1 2 1 21 2 1 2

2 1

2 1 1 1 2 2 1 1 2 2

( ) ( )

1 1

2 2

2 4 2 2 4

G Guv G



d y d v

n d x d y n d u d v

d x d u d y d v

n M G m m n n M G m

∉

∉ ∉ ∉ ∉

∉ ∉ ∉ ∉

= + + +

− −

= + + + +

∑ ∑

∑ ∑ ∑ ∑

∑ ∑ ∑ ∑

( )( ) ( )( ) ( ) ( )( )

1 1

1 1 1 1 1 2 2 2 1 2

2

4( 2 1 2 2 1 2

m n

m n m M G m n m M G

+

− − + − − + −

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

1 2 1 21 2

1 2 1 21 2

1 2 1 2

1 2 2 21 2 1 2

6, ,

2 1, ,

2 1

2 1, , , ,

, ,

[ 2

2 by Lemma 2.1



G G G G


C d x u d y v

n d x n d u d x d u

n d y n d v d y d v

n d x d y n d u d v

⊕ ⊕∉ = ∉ =

∉ = ∉ =

∉ = ∉ = ∉ = ∉ =

= +

= + −

+ + − = + + +

−

∑ ∑

∑ ∑

∑ ∑ ∑ ∑

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

1 2 1 21 2 1 2

1 2 2 21 2 1 2

1 21 2

,, , ,

2 1, , , ,

, ,

2 2

1 1

2

G u v G G Gxy G x y uv G xy G x y uv G u v


G Gxy G x y uv G u v

d x d u d y d v

n d x d y n d u d v

d x d u

=∉ = ∉ ∉ = ∉ =

∉ = ∉ = ∉ = ∉ =

∉ = ∉ =

−

= + + +

− −

∑ ∑ ∑ ∑

∑ ∑ ∑ ∑

∑ ∑ ( ) ( )1 21 2, ,

2 22 1 1 2 1 2

2

4 4 16

G Gxy G x y uv G u v

d y d v

n m n m m m∉ = ∉ =

= + −

∑ ∑

( )

( ) ( )( ) ( )( )

( ) ( ) ( )( ) ( )( )( )( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )( )( ) ( )( )( )

2*1 2

2 2 1 1 1 1 1 2 1 11 2 1 2

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

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

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

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

1 2 2 4 2 21

4 2 1 2 2 2

2 2 4 4 2 1 2

4 4 4

2 2 4 2 2 4 2

DD G G

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


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

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


⊕ ≤ + + + −− − + − + +

+ + − − + −

+ + −

+ + + + + +

( ) ( )( ) ( ) ( )( )

( )( )1 2 1 2

1 1 1 1 1 2 2 2 1 2

12 22 1 1 2 1 2

4 2 1 2 2 1 2

2 4 4 16n n n n

m n m M G m n m M G

n m n m m m−

− − + − − + −

− + −

Lemma 6.2.

[ ] ( )( )1

* 21 2 2m m

mDD K K m−

⊕ = −

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OALib Journal

Proof: Clearly the graph 1KKm ⊕ is the complete graph mK

[ ] ( )( )1

* * 21 2 2m m

m mDD K K DD K m−

⊕ = = − (7)

Remark 6.3. Let 1 mG K= and 2 1G K= . We get

( )1 2 1 2 1 2

1, 1, , 0, 0, 0

2m m

n m n m m m m−

= = = = = =

( ) ( ) ( ) ( ) ( )21 1 1 1 2 1 11 , 0mM G M K m m M G M K= = − = =

( ) ( ) ( ) ( )1 1 1 1 2 1 10, 0mM G M K M G M K= = = = ( ) ( ) ( ) ( )1 1 1 1 2 1 10, 0mM G M K M G M K= = = =

∴ In Theorem 6.1, put 1 mG K= and 2 1G K= , we get

[ ] ( )( )1

* 21 2 2m m

mDD K K m−

⊕ ≤ − (8)


References [1] Dobrynin, A.A. and Kochetova, A.A. (1994) Degree Distance of a Graph: A Degree

Analogue of the Wiener Index. Journal of Chemical Information and Computer Sciences, 34, 1082-1086. https://doi.org/10.1021/ci00021a008

[2] Gutman, I. (1994) Selected Properties of the Schultz Molecular Topological Index. Journal of Chemical Information and Computer Sciences, 34, 1087-1089.

[3] Gutman, I. and Trinajstic, N. (1972) Graph Theory and Moleclar Orbitals. Total pi-Electron Energy of Alternant Hydrocarbons. Chemical Physics Letters, 17, 535- 538.

[4] Devillers, J. and Balaban, A.T. (1999) Topological Indices and Related Descriptors in QSAR and QSPR. Gordon and Breach, Amsterdam, The Netherlands.

[5] Doslic, T. (2008) Vertex-Weighted Wiener Polynomials for Composite Graphs. Ars Mathematica Contemporanea, 1, 66-80.

[6] Ashrafi, A.R., Doslic, T. and Hamzeha, A. (2010) The Zagreb Coindices of Graph Operations. Discrete Applied Mathematics, 158, 1571-1578. https://doi.org/10.1016/j.dam.2010.05.017

[7] Ashrafi, A.R., Doslic, T. and Hamzeha, A. (2011) Extremal Graphs with Respect to The Zagreb Coindices, MATCH Commun. Math. Comput. Chem., 65, 85-92.

[8] Hamzeha, A., Iranmanesh, A., Hossein-Zadeh, S. and Diudea, M.V. (2012) Genera-lized Degree Distance of Trees, Unicyclic and Bicyclic Graphs. Studia Ubb Chemia, LVII, 4, 73-85.

[9] Hamzeha, A., Iranmanesh, A. and Hossein-Zadeh, S. (2013) Some Results on Ge-neralized Degree Distance. Open Journal of Discrete Mathematics, 3, 143-150. https://doi.org/10.4236/ojdm.2013.33026

[10] Imrich, W. and Klavzar, S. (2000) Product Graphs: Structure and Recognition. John Wiley and Sons, New York.

[11] Kahlifeh, M.H., Yousefi-Azari, H. and Ashrafi, A.R. (2008) The Hyper-Wiener In-dex of Graph Operations. Computers and Mathematics with Applications, 56, 1402- 1407. https://doi.org/10.1016/j.camwa.2008.03.003

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https://doi.org/10.1021/ci00021a008https://doi.org/10.1016/j.dam.2010.05.017https://doi.org/10.4236/ojdm.2013.33026https://doi.org/10.1016/j.camwa.2008.03.003

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Sharp Upper Bounds for Multiplicative Degree Distance of Graph OperationsAbstractSubject AreasKeywords1. Introduction2. The Multiplicative Degree Distance of Graph Operations3. The Multiplicative Degree Distance of Composition of Graph4. The Multiplicative Degree Distance of Join of Graph5. The Multiplicative Degree Distance of Disjunction of Graph6. The Multiplicative Degree Distance of Symmetric difference of Graph References

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Scientific Research PublishingOpen Access Library Journal 2017, Volume 4, e2987 ISSN Online: 2333-9721 ISSN Print: 2333-9705 . Sharp Upper Bounds for Multiplicative Degree Distance