﻿ Sharp Upper Bounds for Multiplicative Degree Distance of Graph Operations

Open Access Library Journal
Vol.04 No.07(2017), Article ID:77523,18 pages
10.4236/oalib.1102987

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    Received: August 18, 2016; Accepted: July 8, 2017; Published: July 11, 2017

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 $V\left(G\right)$ and $E\left(G\right)$ , respectively. ${d}_{G}\left(v\right)$ 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 $v\left(G\right)$ . The number of elements in the edge set of a graph G is called the size of G and is denoted by $e\left(G\right)$ . A graph with order n and size m edges is called a $\left(n,m\right)$ -graph. For any $u,v\in V\left(G\right)$ , the distance between u and v in G, denoted by ${d}_{G}\left(u,v\right)$ , is the length of a shortest $\left(u,v\right)$ -path in G. The edge connective the vertices u and v will be denoted by uv. The complement $\stackrel{¯}{G}$ of the graph G is the graph with vertex set $V\left(G\right)$ , in which two vertices in $\stackrel{¯}{G}$ are adjacent if and only if they are not adjacent in G.

The join of graphs ${G}_{1}$ and ${G}_{2}$ is denoted by ${G}_{1}+{G}_{2}$ , and it is the graph with vertex set $V\left({G}_{1}\right)\cup V\left({G}_{2}\right)$ and the edge set $E\left({G}_{1}+{G}_{2}\right)=E\left({G}_{1}\right)\cup E\left({G}_{2}\right)$ $\cup \left\{{u}_{1}{u}_{2}|{u}_{1}\in V\left({G}_{1}\right),{u}_{2}\in V\left({G}_{2}\right)\right\}.$ The composition of graphs ${G}_{1}$ and ${G}_{2}$ is denoted by ${G}_{1}\left[{G}_{2}\right]$ , and it is the graph with vertex set $V\left({G}_{1}\right)×V\left({G}_{2}\right)$ , and two vertices $u=\left({u}_{1},{u}_{2}\right)$ and $v=\left({v}_{1},{v}_{2}\right)$ are adjacent if ( ${u}_{1}$ is adjacent to ${v}_{1}$ ) or ( ${u}_{1}={v}_{1}$ and ${u}_{2}$ and ${v}_{2}$ are adjacent). The disjunction of graphs ${G}_{1}$ and ${G}_{2}$ is denoted by ${G}_{1}\vee {G}_{2}$ , and it is the graph with vertex set $V\left({G}_{1}\right)×V\left({G}_{2}\right)$ and $E\left({G}_{1}\vee {G}_{2}\right)=\left\{\left({u}_{1},{u}_{2}\right)\left({v}_{1},{v}_{2}\right)|{u}_{1}{v}_{1}\in E\left({G}_{1}\right)\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}{u}_{2}{v}_{2}\in E\left({G}_{2}\right)\right\}.$ The symmetric di- fference of graphs ${G}_{1}$ and ${G}_{2}$ is denoted by ${G}_{1}\oplus {G}_{2}$ , and it is the graph with vertex set $V\left({G}_{1}\right)×V\left({G}_{2}\right)$ and edge set $E\left({G}_{1}\oplus {G}_{2}\right)=\left\{\left({u}_{1},{u}_{2}\right)\left({v}_{1},{v}_{2}\right)\right)|{u}_{1}{v}_{1}\in E\left({G}_{1}\right)\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}{u}_{2}{v}_{2}\in E\left({G}_{2}\right)\text{\hspace{0.17em}}\text{butnotboth}\right\}.$

Let G be a connected graph. The Wiener index $W\left(G\right)$ of a graph G is defined as

$W\left(G\right)=\underset{\left\{u,v\right\}\subseteq V\left(G\right)}{\sum }{d}_{G}\left(u,v\right)=\frac{1}{2}\underset{u,v\in V\left(G\right)}{\sum }{d}_{G}\left(u,v\right).$

Dobrynin and Kochetova  and Gutman  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

$DD\left(G\right)=\underset{\left\{u,v\right\}\subseteq V\left(G\right)}{\sum }{d}_{G}\left(u,v\right)\left[{d}_{G}\left(u\right)+{d}_{G}\left(v\right)\right]=\frac{1}{2}\underset{u,v\in V\left(G\right)}{\sum }{d}_{G}\left(u,v\right)\left[{d}_{G}\left(u\right)+{d}_{G}\left(v\right)\right].$

The Zagreb indices have been introduced more than thirth years ago by Gutman and Trianjestic  . The first Zagreb index ${M}_{1}\left(G\right)$ of a graph G is defined as

${M}_{1}\left(G\right)=\underset{uv\in E\left(G\right)}{\sum }\left[{d}_{G}\left(u\right)+{d}_{G}\left(v\right)\right]=\underset{v\in V\left(G\right)}{\sum }{d}_{G}^{2}\left(v\right).$

The second Zagreb index ${M}_{2}\left(G\right)$ of a graph G is defined as

${M}_{2}\left(G\right)=\underset{uv\in E\left(G\right)}{\sum }{d}_{G}\left(u\right){d}_{G}\left(v\right).$

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

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

${\stackrel{¯}{M}}_{1}\left(G\right)=\underset{uv\notin E\left(G\right)}{\sum }\left[{d}_{G}\left(u\right)+{d}_{G}\left(v\right)\right]$

The second Zagreb coindex of a graph G is

${\stackrel{¯}{M}}_{2}\left(G\right)=\underset{uv\notin E\left(G\right)}{\sum }{d}_{G}\left(u\right){d}_{G}\left(v\right),$

respectively.

In  , Hamzeh, Iranmanesh Hossein-Zadeh and M.V. Diudea recently 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 $x=1$ , see  .

In this paper, we defne a new graph invariant named multiplicative version of degree distance of a graph denoted by $D{D}^{*}\left(G\right)$ and defined by

${\left[D{D}^{*}\left(G\right)\right]}^{2}=\underset{u,v\in V\left(G\right),u\ne v}{\prod }{d}_{G}\left(u,v\right)\left[{d}_{G}\left(u\right)+{d}_{G}\left(v\right)\right].$

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.   , Let ${G}_{1}$ and ${G}_{2}$ be two simple connected graphs. The number of vertices and edges of graph ${G}_{i}$ is denoted by ${n}_{i}$ and ${e}_{i}$ respectively for $i=1,2$ . Then we have

1. ${d}_{{G}_{1}+{G}_{2}}\left(u,v\right)=\left(\begin{array}{cc}1,& uv\in E\left({G}_{1}\right)\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}uv\in E\left({G}_{2}\right)\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}\left(u\in V\left({G}_{1}\right)\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}v\in V\left({G}_{2}\right)\right)\\ 2,& \text{otherwise}\end{array}$

For a vertex u of ${G}_{1}$ , ${d}_{{G}_{1}+{G}_{2}}\left(u\right)={d}_{{G}_{1}}\left(u\right)+{n}_{2},$ and for a vertex v of ${G}_{2}$ , ${d}_{{G}_{1}+{G}_{2}}\left(v\right)={d}_{{G}_{2}}\left(v\right)+{n}_{1}.$

2. ${d}_{{G}_{1}\left[{G}_{2}\right]}\left(\left({u}_{1},{v}_{1}\right),\left({u}_{2},{v}_{2}\right)\right)=\left(\begin{array}{ll}{d}_{{G}_{1}}\left({u}_{1},{u}_{2}\right),\hfill & \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{u}_{1}\ne {u}_{2}\hfill \\ 1,\hfill & {u}_{1}={u}_{2},{v}_{1}{v}_{2}\in E\left({G}_{2}\right)\hfill \\ 2,\hfill & \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{otherwise}\hfill \end{array}$

${d}_{{G}_{1}\left[{G}_{2}\right]}\left(u,v\right)={n}_{2}{d}_{{G}_{1}}\left(u\right)+{d}_{{G}_{2}}\left(v\right).$

3. ${d}_{{G}_{1}\vee {G}_{2}}\left(\left({u}_{1},{v}_{1}\right),\left({u}_{2},{v}_{2}\right)\right)=\left(\begin{array}{cc}1,& {u}_{1}{u}_{2}\in E\left({G}_{1}\right)\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}{v}_{1}{v}_{2}\in E\left({G}_{2}\right)\\ 2,& \text{otherwise}\end{array}$

${d}_{{G}_{1}\vee {G}_{2}}\left(\left(u,v\right)\right)={n}_{2}{d}_{{G}_{1}}\left(u\right)+{n}_{1}{d}_{{G}_{2}}\left(v\right)-{d}_{{G}_{1}}\left(u\right){d}_{{G}_{2}}\left(v\right).$

4. ${d}_{{G}_{1}\oplus {G}_{2}}\left(\left({u}_{1},{v}_{1}\right),\left({u}_{2},{v}_{2}\right)\right)=\left(\begin{array}{cc}1,& {u}_{1}{u}_{2}\in E\left({G}_{1}\right)\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}{v}_{1}{v}_{2}\in E\left({G}_{2}\right)\text{\hspace{0.17em}}\text{butnotboth}\\ 2,& \text{otherwise}\end{array}$

${d}_{{G}_{1}\oplus {G}_{2}}\left(\left(u,v\right)\right)={n}_{2}{d}_{{G}_{1}}\left(u\right)+{n}_{1}{d}_{{G}_{2}}\left(v\right)-2{d}_{{G}_{1}}\left(u\right){d}_{{G}_{2}}\left(v\right).$

Lemma 2.2. (Arithmetic Geometric inequality)

Let ${x}_{1},{x}_{2},\cdots ,{x}_{n}$ be non-negative numbers. Then $\frac{{x}_{1}+{x}_{2}+\cdots +{x}_{n}}{n}\ge \sqrt[n]{{x}_{1}{x}_{2}\cdots {x}_{n}}$

Remark 2.3. For a graph G, let $A\left(G\right)$ = { $\left(x,y\right)\in V\left(G\right)×V\left(G\right)|x$ and $y$ are adjacent in $G$ } and let $B\left(G\right)$ = { $\left(x,y\right)\in V\left(G\right)×V\left(G\right)|x$ and $y$ are not adjacent in $G$ }. For each $x\in V\left(G\right),\left(x,x\right)\in B\left(G\right)$ . Clearly, $A\left(G\right)\cup B\left(G\right)=V\left(G\right)×V\left(G\right).$ Let $C\left(G\right)=\left\{\left(x,x\right)|x\in V\left(G\right)\right\}$ and $D\left(G\right)=B\left(G\right)-C\left(G\right).$ Clearly $B\left(G\right)=C\left(G\right)\cup D\left(G\right)$ , $C\left(G\right)\cap D\left(G\right)=\varnothing$ .

The summation $\underset{\left(x,y\right)\in A\left(G\right)}{\sum }$ runs over the ordered pairs of $A\left(G\right)$ . For simplicity, we write the summation $\underset{\left(x,y\right)\in A\left(G\right)}{\sum }$ as $\underset{xy\in G}{\sum }$ . Similarly, we write the summation $\underset{\left(x,y\right)\in B\left(G\right)}{\sum }$ as $\underset{xy\notin G}{\sum }$ . Also the summation $\underset{xy\in E\left(G\right)}{\sum }$ runs over the edges of G. We denote the summation $\underset{x,y\in V\left(G\right)}{\sum }$ by $\underset{x,y\in G}{\sum }$ and similarly $\underset{x,y\in V\left(G\right)}{\sum }$ by $\underset{x,y\in G}{\prod }$ . The summation $\underset{\left(x,y\right)\in D\left(G\right)}{\sum }$ eqivalent to $\underset{xy\notin G,x\ne y}{\sum }$ and similarly the summation $\underset{\left(x,y\right)\in C\left(G\right)}{\sum }$ eqivalent to $\underset{xy\notin G,x=y}{\sum }$ .

Lemma 2.4. Let G be a graph. Then

$\underset{xy\in G}{\sum }1=2e\left( G \right)$

Proof:

$\underset{xy\in G}{\sum }1=2\underset{xy\in E\left(G\right)}{\sum }1=2e\left( G \right)$

Lemma 2.5.

$\underset{xy\in G}{\sum }{d}_{G}\left(x\right)={M}_{1}\left( G \right)$

Proof: Let $x\in V\left(G\right)$ and $t={d}_{G}\left(x\right)$ . Let ${y}_{1},{y}_{2},\cdots ,{y}_{t}$ be the neighbours of x. Each orderd pair $\left(x,{y}_{i}\right),\text{ }1\le i\le t,$ contributes ${d}_{G}\left(x\right)$ to the sum. Thus these orderd pairs contribute ${d}_{G}^{2}\left(x\right)$ to the sum. Hence

$\underset{xy\in G}{\sum }{d}_{G}\left(x\right)=\underset{x\in V\left(G\right)}{\sum }{d}_{G}^{2}\left(x\right)={M}_{1}\left( G \right)$

Lemma 2.6.

$\underset{xy\in G}{\sum }{d}_{G}\left(x\right){d}_{G}\left(y\right)=2{M}_{2}\left( G \right)$

Proof: Clearly,

$\underset{xy\in G}{\sum }{d}_{G}\left(x\right){d}_{G}\left(y\right)=2\underset{xy\in E\left(G\right)}{\sum }{d}_{G}\left(x\right){d}_{G}\left(y\right)=2{M}_{2}\left(G\right).$

Lemma 2.7.

$\underset{xy\notin G}{\sum }1=2e\left(\stackrel{¯}{G}\right)+v\left( G \right)$

Proof:

$\underset{xy\notin G}{\sum }1=\underset{\left(x,y\right)\in D\left(G\right)}{\sum }1+\underset{\left(x,x\right)\in C\left(G\right)}{\sum }1=2e\left(\stackrel{¯}{G}\right)+v\left( G \right)$

Lemma 2.8.

$\underset{xy\notin G}{\sum }{d}_{G}\left(x\right)=2e\left(\stackrel{¯}{G}\right)\left(v\left(G\right)-1\right)+2e\left(G\right)-{M}_{1}\left( G ¯ \right)$

Proof.

$\begin{array}{c}\underset{xy\notin G}{\sum }{d}_{G}\left(x\right)=\underset{\left(x,y\right)\in D\left(G\right)}{\sum }{d}_{G}\left(x\right)+\underset{\left(x,x\right)\in C\left(G\right)}{\sum }{d}_{G}\left(x\right)\\ =\underset{\left(x,y\right)\in D\left(G\right)}{\sum }\left\{v\left(G\right)-1-{d}_{\stackrel{¯}{G}}\left(x\right)\right\}+\underset{\left(x,x\right)\in C\left(G\right)}{\sum }{d}_{G}\left(x\right)\\ =\left(v\left(G\right)-1\right)\underset{\left(x,y\right)\in D\left(G\right)}{\sum }1-\underset{\left(x,y\right)\in D\left(G\right)}{\sum }{d}_{\stackrel{¯}{G}}\left(x\right)+2e\left(G\right)\\ =\left(v\left(G\right)-1\right)2e\left(\stackrel{¯}{G}\right)-\underset{\left(x,y\right)\in A\left(\stackrel{¯}{G}\right)}{\sum }{d}_{\stackrel{¯}{G}}^{2}\left(x\right)+2e\left(G\right)\\ =\left(v\left(G\right)-1\right)2e\left(\stackrel{¯}{G}\right)-\underset{xy\in \stackrel{¯}{G}}{\sum }{d}_{\stackrel{¯}{G}}^{2}\left(x\right)+2e\left(G\right)\\ =2e\left(\stackrel{¯}{G}\right)\left(v\left(G\right)-1\right)+2e\left(G\right)-{M}_{1}\left(\stackrel{¯}{G}\right)\text{ }\text{\hspace{0.17em}}\text{ }\text{ }\text{by}\text{\hspace{0.17em}}\text{Lemma}\text{ }\text{ }\text{ }2.5\end{array}$

Lemma 2.9.

$\underset{xy\notin G}{\sum }{d}_{G}\left(x\right){d}_{G}\left(y\right)=2{\stackrel{¯}{M}}_{2}\left(G\right)+{M}_{1}\left( G \right)$

Proof:

$\begin{array}{c}\underset{xy\notin G}{\sum }{d}_{G}\left(x\right){d}_{G}\left(y\right)=\underset{\left(x,y\right)\in D\left(G\right)}{\sum }{d}_{G}\left(x\right){d}_{G}\left(y\right)+\underset{\left(x,x\right)\in C\left(G\right)}{\sum }{d}_{G}\left(x\right){d}_{G}\left(x\right)\\ =2\underset{xy\notin E\left(G\right)}{\sum }{d}_{G}\left(x\right){d}_{G}\left(y\right)+\underset{x\in V\left(G\right)}{\sum }{d}_{G}^{2}\left(x\right)\\ =2{\stackrel{¯}{M}}_{2}\left(G\right)+{M}_{1}\left( G \right)\end{array}$

Lemma 2.10.

$\underset{xy\notin G}{\sum }\left[{d}_{G}\left(x\right)+{d}_{G}\left(y\right)\right]=2{\stackrel{¯}{M}}_{1}\left(G\right)+4e\left( G \right)$

Proof:

$\begin{array}{c}\underset{xy\notin G}{\sum }\left[{d}_{G}\left(x\right)+{d}_{G}\left(y\right)\right]=\underset{\left(x,y\right)\in C\left(G\right)}{\sum }\left[{d}_{G}\left(x\right)+{d}_{G}\left(y\right)\right]+\underset{\left(x,y\right)\in D\left(G\right)}{\sum }\left[{d}_{G}\left(x\right)+{d}_{G}\left(y\right)\right]\\ =\underset{x\in V\left(G\right)}{\sum }2{d}_{G}\left(x\right)+2\underset{xy\notin E\left(G\right)}{\sum }\left[{d}_{G}\left(x\right)+{d}_{G}\left(y\right)\right]\\ =4e\left(G\right)+2{\stackrel{¯}{M}}_{1}\left( G \right)\end{array}$

3. The Multiplicative Degree Distance of Composition of Graph

Theorem 3.1. Let ${G}_{i},i=1,2$ , be a $\left({n}_{i},{m}_{i}\right)$ -graph. Then

$\begin{array}{l}{\left[D{D}^{*}\left({G}_{1}\left[{G}_{2}\right]\right)\right]}^{2}\\ \le \left\{\frac{1}{{n}_{1}{n}_{2}\left({n}_{1}{n}_{2}-1\right)}\left[4{M}_{1}\left({G}_{2}\right)W\left({G}_{1}\right)+4{n}_{2}{m}_{2}DD\left({G}_{1}\right)+4{n}_{1}{\stackrel{¯}{M}}_{1}\left({G}_{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+8{n}_{2}^{2}{m}_{1}\left({n}_{2}-1\right)+2{n}_{1}{M}_{1}\left({G}_{2}\right)+8{m}_{1}{n}_{2}{m}_{2}+4W\left({G}_{1}\right){\stackrel{¯}{M}}_{1}\left({G}_{2}\right)\\ \text{ }{\begin{array}{c}\\ \end{array}+2{n}_{2}DD\left({G}_{1}\right)\left(2{\stackrel{¯}{m}}_{2}+{n}_{2}\right)\right]\right\}}^{{n}_{1}{n}_{2}\left({n}_{1}{n}_{2}-1\right)}\end{array}$

Proof: ${\left[D{D}^{*}{G}_{1}\left[{G}_{2}\right]\right]}^{2}\le {\left\{\frac{1}{{n}_{1}{n}_{2}\left({n}_{1}{n}_{2}-1\right)}\left({S}_{3}+{S}_{1}+{S}_{2}+{S}_{4}\right)\right\}}^{{n}_{1}{n}_{2}\left({n}_{1}{n}_{2}-1\right)}$

where ${S}_{3},{S}_{1},{S}_{2},{S}_{4}$ are terms of the above sums taken in order.

Next we calculate ${S}_{1},{S}_{2},{S}_{3}$ and ${S}_{4}$ separately.

$\begin{array}{c}{S}_{1}=\underset{x,y\in {G}_{1},x\ne y}{\sum }\text{ }\text{ }\text{ }\text{ }\underset{uv\in {G}_{2}}{\sum }{d}_{{G}_{1}\left[{G}_{2}\right]}\left(\left(x,u\right),\left(y,v\right)\right)\left[{d}_{{G}_{1}\left[{G}_{2}\right]}\left(x,u\right)+{d}_{{G}_{1}\left[{G}_{2}\right]}\left(y,v\right)\right]\\ =\underset{x,y\in {G}_{1},x\ne y}{\sum }\text{ }\text{ }\text{ }\text{ }\underset{uv\in {G}_{2}}{\sum }{d}_{{G}_{1}}\left(x,y\right)\left[{d}_{{G}_{2}}\left(u\right)+{n}_{2}{d}_{{G}_{1}}\left(x\right)+{d}_{{G}_{2}}\left(v\right)+{n}_{2}{d}_{{G}_{1}}\left(y\right)\right]\text{ }\text{ }\text{ }\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{Lemma}\text{\hspace{0.17em}}\text{2}.1\\ ={n}_{2}\underset{x,y\in {G}_{1},x\ne y}{\sum }{d}_{{G}_{1}}\left(x,y\right)\left[{d}_{{G}_{1}}\left(x\right)+{d}_{{G}_{1}}\left(y\right)\right]\underset{uv\in {G}_{2}}{\sum }1+\underset{x,y\in {G}_{1},x\ne y}{\sum }{d}_{{G}_{1}}\left(x,y\right)\underset{uv\in {G}_{2}}{\sum }\left[{d}_{{G}_{2}}\left(u\right)+{d}_{{G}_{2}}\left(v\right)\right]\\ =4{n}_{2}{m}_{2}DD\left({G}_{1}\right)+4{M}_{1}\left({G}_{2}\right)W\left( G 1 \right)\end{array}$

$\begin{array}{c}{S}_{2}=\underset{x,y\in {G}_{1},x=y}{\sum }\text{ }\text{ }\text{ }\text{ }\underset{uv\notin {G}_{2}}{\sum }{d}_{{G}_{1}\left[{G}_{2}\right]}\left(\left(x,u\right),\left(y,v\right)\right)\left[{d}_{{G}_{1}\left[{G}_{2}\right]}\left(x,u\right)+{d}_{{G}_{1}\left[{G}_{2}\right]}\left(y,v\right)\right]\\ =\underset{x,y\in {G}_{1},x=y}{\sum }\left\{\underset{uv\notin {G}_{2},u=v}{\sum }{d}_{{G}_{1}\left[{G}_{2}\right]}\left(\left(x,u\right),\left(y,v\right)\right)\left[{d}_{{G}_{1}\left[{G}_{2}\right]}\left(x,u\right)+{d}_{{G}_{1}\left[{G}_{2}\right]}\left(y,v\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{uv\notin {G}_{2},u\ne v}{\sum }{d}_{{G}_{1}\left[{G}_{2}\right]}\left(\left(x,u\right),\left(y,v\right)\right)\left[{d}_{{G}_{1}\left[{G}_{2}\right]}\left(x,u\right)+{d}_{{G}_{1}\left[{G}_{2}\right]}\left(y,v\right)\right]\right\}\\ =0+\underset{x,y\in {G}_{1},x=y}{\sum }\text{ }\text{ }\text{ }\text{ }\underset{uv\notin {G}_{2},u\ne v}{\sum }{d}_{{G}_{1}\left[{G}_{2}\right]}\left(\left(x,u\right),\left(y,v\right)\right)\left[{d}_{{G}_{1}\left[{G}_{2}\right]}\left(x,u\right)+{d}_{{G}_{1}\left[{G}_{2}\right]}\left(y,v\right)\right]\\ =\underset{x,y\in {G}_{1},x=y}{\sum }\text{ }\text{ }\text{ }\text{ }\underset{uv\notin {G}_{2},u\ne v}{\sum }{d}_{{G}_{1}}\left(x,y\right)\left[{d}_{{G}_{2}}\left(u\right)+{n}_{2}{d}_{{G}_{1}}\left(x\right)+{d}_{{G}_{2}}\left(v\right)+{n}_{2}{d}_{{G}_{1}}\left(y\right)\right]\text{ }\text{ }\text{ }\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{Lemma}\text{\hspace{0.17em}}\text{2}\text{.1}\text{ }\\ =2\underset{uv\notin {G}_{2},u\ne v}{\sum }\left[{d}_{{G}_{2}}\left(u\right)+{d}_{{G}_{2}}\left(v\right)\right]\underset{x,y\in {G}_{1},x=y}{\sum }1+2{n}_{2}\left(\underset{uv\notin {G}_{2},u\ne v}{\sum }1\right)\underset{x,y\in {G}_{1},x=y}{\sum }\left[{d}_{{G}_{1}}\left(x\right)+{d}_{{G}_{1}}\left(y\right)\right]\\ =4{n}_{1}{\stackrel{¯}{M}}_{1}\left({G}_{2}\right)+8{n}_{2}{m}_{1}{n}_{2}\left({n}_{2}-1\right)\end{array}$

$\begin{array}{c}{S}_{3}=\underset{x,y\in {G}_{1},x=y}{\sum }\text{ }\text{ }\text{ }\text{ }\underset{uv\in {G}_{2}}{\sum }{d}_{{G}_{1}\left[{G}_{2}\right]}\left(\left(x,u\right),\left(y,v\right)\right)\left[{d}_{{G}_{1}\left[{G}_{2}\right]}\left(x,u\right)+{d}_{{G}_{1}\left[{G}_{2}\right]}\left(y,v\right)\right]\\ =1\cdot \underset{x,y\in {G}_{1},x=y}{\sum }\text{ }\text{ }\text{ }\text{ }\underset{uv\in {G}_{2}}{\sum }\left[{d}_{{G}_{1}\left[{G}_{2}\right]}\left(x,u\right)+{d}_{{G}_{1}\left[{G}_{2}\right]}\left(y,v\right)\right]\\ =\underset{x,y\in {G}_{1},x=y}{\sum }\text{ }\text{ }\text{ }\text{ }\underset{uv\in {G}_{2}}{\sum }\left[{d}_{{G}_{2}}\left(u\right)+{d}_{{G}_{1}}\left(x\right){n}_{2}+{d}_{{G}_{2}}\left(v\right)+{n}_{2}{d}_{{G}_{1}}\left(y\right)\right]\text{ }\text{ }\text{ }\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{Lemma}\text{\hspace{0.17em}}\text{2}\text{.1}\text{ }\\ =\left(\underset{x,y\in {G}_{1},x=y}{\sum }1\right)\underset{uv\in {G}_{2}}{\sum }\left[{d}_{{G}_{2}}\left(u\right)+{d}_{{G}_{2}}\left(v\right)\right]+{n}_{2}\underset{x,y\in {G}_{1},x=y}{\sum }\left[{d}_{{G}_{1}}\left(x\right)+{d}_{{G}_{1}}\left(y\right)\right]\left(\underset{uv\in {G}_{2}}{\sum }1\right)\\ =2{n}_{1}{M}_{1}\left({G}_{2}\right)+8{n}_{2}{m}_{1}{m}_{2}\end{array}$

$\begin{array}{c}{S}_{4}=\underset{x,y\in {G}_{1},x\ne y}{\sum }\text{ }\text{ }\text{ }\text{ }\underset{uv\notin {G}_{2}}{\sum }{d}_{{G}_{1}\left[{G}_{2}\right]}\left(\left(x,u\right),\left(y,v\right)\right)\left[{d}_{{G}_{1}\left[{G}_{2}\right]}\left(x,u\right)+{d}_{{G}_{1}\left[{G}_{2}\right]}\left(y,v\right)\right]\\ =\underset{x,y\in {G}_{1},x\ne y}{\sum }\text{ }\text{ }\text{ }\text{ }\underset{uv\notin {G}_{2}}{\sum }{d}_{{G}_{1}}\left(x,y\right)\left[{d}_{{G}_{1}\left[{G}_{2}\right]}\left(x,u\right)+{d}_{{G}_{1}\left[{G}_{2}\right]}\left(y,v\right)\right]\\ =\underset{x,y\in {G}_{1},x\ne y}{\sum }\text{ }\text{ }\text{ }\text{ }\underset{uv\notin {G}_{2}}{\sum }{d}_{{G}_{1}}\left(x,y\right)\left[{d}_{{G}_{2}}\left(u\right)+{d}_{{G}_{1}}\left(x\right){n}_{2}+{d}_{{G}_{2}}\left(v\right)+{n}_{2}{d}_{{G}_{1}}\left(y\right)\right]\text{ }\text{ }\text{ }\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{Lemma}\text{\hspace{0.17em}}\text{2}\text{.1}\text{ }\\ =\left(\underset{x,y\in {G}_{1},x\ne y}{\sum }{d}_{{G}_{1}}\left(x,y\right)\right)\underset{uv\notin {G}_{2}}{\sum }\left[{d}_{{G}_{2}}\left(u\right)+{d}_{{G}_{2}}\left(v\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{n}_{2}\underset{x,y\in {G}_{1},x\ne y}{\sum }{d}_{{G}_{1}}\left(x,y\right)\left[{d}_{{G}_{1}}\left(x\right)+{d}_{{G}_{1}}\left(y\right)\right]\left(\underset{uv\notin {G}_{2}}{\sum }1\right)\\ =4W\left({G}_{1}\right){\stackrel{¯}{M}}_{1}\left({G}_{2}\right)+2{n}_{2}DD\left({G}_{1}\right)\left(2{\stackrel{¯}{m}}_{2}+{n}_{2}\right)\end{array}$

$\begin{array}{c}{\left[D{D}^{*}\left({G}_{1}\left[{G}_{2}\right]\right)\right]}^{2}\le \left\{\frac{1}{{n}_{1}{n}_{2}\left({n}_{1}{n}_{2}-1\right)}\left[4{M}_{1}\left({G}_{2}\right)W\left({G}_{1}\right)+4{n}_{2}{m}_{2}DD\left({G}_{1}\right)+4{n}_{1}{\stackrel{¯}{M}}_{1}\left({G}_{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+8{n}_{2}^{2}{m}_{1}\left({n}_{2}-1\right)+2{n}_{1}{M}_{1}\left({G}_{2}\right)+8{m}_{1}{n}_{2}{m}_{2}+4W\left({G}_{1}\right){\stackrel{¯}{M}}_{1}\left({G}_{2}\right)\\ \text{ }{\begin{array}{c}\\ \end{array}+2{n}_{2}DD\left({G}_{1}\right)\left(2{\stackrel{¯}{m}}_{2}+{n}_{2}\right)\right]\right\}}^{{n}_{1}{n}_{2}\left({n}_{1}{n}_{2}-1\right)}\end{array}$

Lemma 3.2.

$D{D}^{*}{K}_{n}\left[{K}_{r}\right]={\left[2\left(nr-1\right)\right]}^{\frac{nr\left(nr-1\right)}{2}}$

Proof: Clearly the graph ${K}_{n}\left[{K}_{r}\right]$ is the complete graph ${K}_{nr}$ .

$\therefore D{D}^{*}\left({K}_{n}\left[{K}_{r}\right]\right)=D{D}^{*}\left({K}_{nr}\right)={\left[2\left(nr-1\right)\right]}^{\frac{nr\left(nr-1\right)}{2}}$ (1)

Remark 3.3. Let ${G}_{1}={K}_{n}$ and ${G}_{2}={K}_{r}$ . We get,

$DD\left({G}_{1}\right)=2\left(n-1\right)\frac{n\left(n-1\right)}{2}=n{\left(n-1\right)}^{2},\text{ }\text{ }{m}_{1}=\frac{n\left(n-1\right)}{2},\text{ }\text{ }\text{ }\text{ }W\left({G}_{1}\right)=\frac{n\left(n-1\right)}{2}$

${M}_{1}\left({G}_{2}\right)=r{\left(r-1\right)}^{2},\text{ }\text{ }{\stackrel{¯}{M}}_{1}\left({G}_{2}\right)=0,\text{\hspace{0.17em}}{\stackrel{¯}{m}}_{2}=0,\text{ }\text{\hspace{0.17em}}{n}_{1}=n,\text{\hspace{0.17em}}{n}_{2}=r,\text{ }\text{\hspace{0.17em}}{m}_{2}=\frac{r\left(r-1\right)}{2}$

$\therefore$ In Theorem 3.1, put ${G}_{1}={K}_{n}$ and ${G}_{2}={K}_{r}$ , we get

$D{D}^{*}{K}_{n}\left[{K}_{r}\right]\le {\left[2\left(nr-1\right)\right]}^{\frac{nr\left(nr-1\right)}{2}}$ (2)

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

4. The Multiplicative Degree Distance of Join of Graph

Theorem 4.1. Let ${G}_{i},i=1,2$ , be a $\left({n}_{i},{m}_{i}\right)$ -graph and let ${\stackrel{¯}{m}}_{i}=e\left({\stackrel{¯}{G}}_{i}\right)$ . Then

$\begin{array}{c}{\left[D{D}^{*}\left({G}_{1}+{G}_{2}\right)\right]}^{2}\le \left\{\frac{1}{\left({n}_{1}+{n}_{2}\right)\left({n}_{1}+{n}_{2}-1\right)}\left[2{M}_{1}\left({G}_{1}\right)+4{n}_{2}{m}_{1}+4{\stackrel{¯}{M}}_{1}\left({G}_{1}\right)+8{n}_{2}{\stackrel{¯}{m}}_{1}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2{m}_{1}{n}_{2}+2{m}_{2}{n}_{1}+{n}_{1}{n}_{2}\left({n}_{1}+{n}_{2}\right)+2{M}_{1}\left({G}_{2}\right)\\ {\begin{array}{c}\\ \end{array}+4{n}_{1}{m}_{2}+4{\stackrel{¯}{M}}_{1}\left({G}_{2}\right)+8{n}_{1}{\stackrel{¯}{m}}_{2}\right]\right\}}^{\left({n}_{1}+{n}_{2}\right)\left({n}_{1}+{n}_{2}-1\right)}\end{array}$

Proof: ${\left[D{D}^{*}\left({G}_{1}+{G}_{2}\right)\right]}^{2}\le {\left[\frac{1}{\left({n}_{1}+{n}_{2}\right)\left({n}_{1}+{n}_{2}-1\right)}\left({J}_{1}+2{J}_{2}+{J}_{3}\right)\right]}^{\left({n}_{1}+{n}_{2}\right)\left({n}_{1}+{n}_{2}-1\right)}$

where ${J}_{1},{J}_{2},{J}_{3}$ are terms of the above sums taken in order.

Next we calculate ${J}_{1},{J}_{2}$ and ${J}_{3}$ separately one by one. Now,

$\begin{array}{c}{J}_{1}=\underset{x\in V\left({G}_{1}\right)}{\sum }\text{ }\text{ }\text{ }\text{ }\underset{y\in V\left({G}_{1}\right)}{\sum }{d}_{\left({G}_{1}+{G}_{2}\right)}\left(x,y\right)\left[{d}_{\left({G}_{1}+{G}_{2}\right)}\left(x\right)+{d}_{\left({G}_{1}+{G}_{2}\right)}\left(y\right)\right]\\ =\underset{x,y\in V\left({G}_{1}\right)}{\sum }{d}_{\left({G}_{1}+{G}_{2}\right)}\left(x,y\right)\left[{d}_{\left({G}_{1}+{G}_{2}\right)}\left(x\right)+{d}_{\left({G}_{1}+{G}_{2}\right)}\left(y\right)\right]\\ =\underset{xy\in {G}_{1}}{\sum }{d}_{\left({G}_{1}+{G}_{2}\right)}\left(x,y\right)\left[{d}_{\left({G}_{1}+{G}_{2}\right)}\left(x\right)+{d}_{\left({G}_{1}+{G}_{2}\right)}\left(y\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{xy\notin {G}_{1},x\ne y}{\sum }{d}_{\left({G}_{1}+{G}_{2}\right)}\left(x,y\right)\left[{d}_{\left({G}_{1}+{G}_{2}\right)}\left(x\right)+{d}_{\left({G}_{1}+{G}_{2}\right)}\left(y\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{xy\notin {G}_{1},x=y}{\sum }{d}_{\left({G}_{1}+{G}_{2}\right)}\left(x,y\right)\left[{d}_{\left({G}_{1}+{G}_{2}\right)}\left(x\right)+{d}_{\left({G}_{1}+{G}_{2}\right)}\left(y\right)\right]\\ =1\cdot \underset{xy\in {G}_{1}}{\sum }\left[{d}_{\left({G}_{1}+{G}_{2}\right)}\left(x\right)+{d}_{\left({G}_{1}+{G}_{2}\right)}\left(y\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\cdot \underset{xy\notin {G}_{1},x\ne y}{\sum }\left[{d}_{\left({G}_{1}+{G}_{2}\right)}\left(x\right)+{d}_{\left({G}_{1}+{G}_{2}\right)}\left(y\right)\right]+0\\ =\underset{xy\in {G}_{1}}{\sum }\left[{d}_{{G}_{1}}\left(x\right)+{n}_{2}+{d}_{{G}_{2}}\left(y\right)+{n}_{2}\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\underset{xy\notin {G}_{1},x\ne y}{\sum }\left[{d}_{{G}_{1}}\left(x\right)+{n}_{2}+{d}_{{G}_{2}}\left(y\right)+{n}_{2}\right]\text{ }\text{ }\text{\hspace{0.17em}}\text{ }\text{by}\text{\hspace{0.17em}}\text{Lemma}\text{\hspace{0.17em}}\text{2}\text{.1}\\ =\underset{xy\in {G}_{1}}{\sum }\left[{d}_{{G}_{1}}\left(x\right)+{d}_{{G}_{2}}\left(y\right)\right]+2{n}_{2}\underset{xy\in {G}_{1}}{\sum }1\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\left\{\underset{xy\notin {G}_{1},x\ne y}{\sum }\left[{d}_{{G}_{1}}\left(x\right)+{d}_{{G}_{2}}\left(y\right)\right]+2{n}_{2}\underset{xy\notin {G}_{1},x\ne y}{\sum }1\right\}\\ =2{M}_{1}\left({G}_{1}\right)+4{n}_{2}{m}_{1}+4{\stackrel{¯}{M}}_{1}\left({G}_{1}\right)+8{n}_{2}{\stackrel{¯}{m}}_{1}\end{array}$

$\begin{array}{c}{J}_{2}=\underset{x\in V\left({G}_{1}\right)}{\sum }\text{ }\text{ }\text{ }\text{ }\underset{y\in V\left({G}_{2}\right)}{\sum }{d}_{\left({G}_{1}+{G}_{2}\right)}\left(x,y\right)\left[{d}_{\left({G}_{1}+{G}_{2}\right)}\left(x\right)+{d}_{\left({G}_{1}+{G}_{2}\right)}\left(y\right)\right]\\ =1\underset{x\in V\left({G}_{1}\right)}{\sum }\text{ }\text{ }\text{ }\text{ }\underset{y\in V\left({G}_{2}\right)}{\sum }\left[{d}_{\left({G}_{1}+{G}_{2}\right)}\left(x\right)+{d}_{\left({G}_{1}+{G}_{2}\right)}\left(y\right)\right]\\ =\underset{x\in V\left({G}_{1}\right)}{\sum }\text{ }\text{ }\text{ }\text{ }\underset{y\in V\left({G}_{2}\right)}{\sum }\left[{d}_{{G}_{1}}\left(x\right)+{n}_{2}+{d}_{{G}_{2}}\left(y\right)+{n}_{1}\right]\text{ }\text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{Lemma}\text{\hspace{0.17em}}\text{2}\text{.1}\text{ }\\ =\underset{x\in V\left({G}_{1}\right)}{\sum }{d}_{{G}_{1}}\left(x\right)\underset{y\in V\left({G}_{2}\right)}{\sum }1+\underset{x\in V\left({G}_{1}\right)}{\sum }1\underset{y\in V\left({G}_{2}\right)}{\sum }{d}_{{G}_{2}}\left(y\right)+\left({n}_{1}+{n}_{2}\right)\underset{x\in V\left({G}_{1}\right)}{\sum }1\underset{y\in V\left({G}_{2}\right)}{\sum }1\\ =2{m}_{1}{n}_{2}+2{m}_{2}{n}_{1}+\left({n}_{1}+{n}_{2}\right){n}_{1}{n}_{2}\end{array}$

$\begin{array}{c}{J}_{3}=\underset{x\in V\left({G}_{2}\right)}{\sum }\text{ }\text{ }\text{ }\text{ }\underset{y\in V\left({G}_{2}\right)}{\sum }{d}_{\left({G}_{1}+{G}_{2}\right)}\left(x,y\right)\left[{d}_{\left({G}_{1}+{G}_{2}\right)}\left(x\right)+{d}_{\left({G}_{1}+{G}_{2}\right)}\left(y\right)\right]\\ =\underset{x,y\in V\left({G}_{2}\right)}{\sum }{d}_{\left({G}_{1}+{G}_{2}\right)}\left(x,y\right)\left[{d}_{\left({G}_{1}+{G}_{2}\right)}\left(x\right)+{d}_{\left({G}_{1}+{G}_{2}\right)}\left(y\right)\right]\\ =\underset{xy\in {G}_{2}}{\sum }{d}_{\left({G}_{1}+{G}_{2}\right)}\left(x,y\right)\left[{d}_{\left({G}_{1}+{G}_{2}\right)}\left(x\right)+{d}_{\left({G}_{1}+{G}_{2}\right)}\left(y\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{xy\notin {G}_{2},x\ne y}{\sum }{d}_{\left({G}_{1}+{G}_{2}\right)}\left(x,y\right)\left[{d}_{\left({G}_{1}+{G}_{2}\right)}\left(x\right)+{d}_{\left({G}_{1}+{G}_{2}\right)}\left(y\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{xy\notin {G}_{2},x=y}{\sum }{d}_{\left({G}_{1}+{G}_{2}\right)}\left(x,y\right)\left[{d}_{\left({G}_{1}+{G}_{2}\right)}\left(x\right)+{d}_{\left({G}_{1}+{G}_{2}\right)}\left(y\right)\right]\end{array}$

$\begin{array}{c}=1\underset{xy\in {G}_{2}}{\sum }\left[{d}_{\left({G}_{1}+{G}_{2}\right)}\left(x\right)+{d}_{\left({G}_{1}+{G}_{2}\right)}\left(y\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\underset{xy\notin {G}_{2},x\ne y}{\sum }\left[{d}_{\left({G}_{1}+{G}_{2}\right)}\left(x\right)+{d}_{\left({G}_{1}+{G}_{2}\right)}\left(y\right)\right]+0\\ =\underset{xy\in {G}_{2}}{\sum }\left[{d}_{{G}_{2}}\left(x\right)+{n}_{1}+{d}_{{G}_{2}}\left(y\right)+{n}_{1}\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\underset{xy\notin {G}_{2},x\ne y}{\sum }\left[{d}_{{G}_{2}}\left(x\right)+{n}_{1}+{d}_{{G}_{2}}\left(y\right)+{n}_{1}\right]\text{ }\text{ }\text{\hspace{0.17em}}\text{ }\text{by}\text{\hspace{0.17em}}\text{Lemma}\text{\hspace{0.17em}}\text{2}\text{.1}\\ =\underset{xy\in {G}_{2}}{\sum }\left[{d}_{{G}_{2}}\left(x\right)+{d}_{{G}_{2}}\left(y\right)\right]+2{n}_{1}\underset{xy\in {G}_{2}}{\sum }1\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\left\{\underset{xy\notin {G}_{2},x\ne y}{\sum }\left[{d}_{{G}_{2}}\left(x\right)+{d}_{{G}_{2}}\left(y\right)\right]+2{n}_{1}\underset{xy\notin {G}_{2},x\ne y}{\sum }1\right\}\\ =2{M}_{1}\left({G}_{2}\right)+4{n}_{1}{m}_{2}+4{\stackrel{¯}{M}}_{1}\left({G}_{2}\right)+8{n}_{1}{\stackrel{¯}{m}}_{2}\end{array}$

$\begin{array}{c}{\left[D{D}^{*}\left({G}_{1}+{G}_{2}\right)\right]}^{2}\le \left\{\frac{1}{\left({n}_{1}+{n}_{2}\right)\left({n}_{1}+{n}_{2}-1\right)}\left[2{M}_{1}\left({G}_{1}\right)+4{n}_{2}{m}_{1}+4{\stackrel{¯}{M}}_{1}\left({G}_{1}\right)+8{n}_{2}{\stackrel{¯}{m}}_{1}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2{m}_{1}{n}_{2}+2{m}_{2}{n}_{1}+{n}_{1}{n}_{2}\left({n}_{1}+{n}_{2}\right)+2{M}_{1}\left({G}_{2}\right)\\ {\begin{array}{c}\\ \end{array}+4{n}_{1}{m}_{2}+4{\stackrel{¯}{M}}_{1}\left({G}_{2}\right)+8{n}_{1}{\stackrel{¯}{m}}_{2}\right]\right\}}^{\left({n}_{1}+{n}_{2}\right)\left({n}_{1}+{n}_{2}-1\right)}\end{array}$

Lemma 4.2.

$\begin{array}{l}D{D}^{*}\left[{K}_{n}+{K}_{r}\right]\\ ={\left[2\left(n+r-1\right)\right]}^{\frac{\left(n+r\right)\left(n+r-1\right)}{2}}\end{array}$

Proof: Clearly the graph ${K}_{n}+{K}_{r}$ is the complete graph ${K}_{n+r}$

$\begin{array}{l}D{D}^{*}\left[{K}_{n}+{K}_{r}\right]\\ =D{D}^{*}\left[{K}_{n+r}\right]\\ ={\left[2\left(n+r-1\right)\right]}^{\frac{\left(n+r\right)\left(n+r-1\right)}{2}}\end{array}$ (3)

Remark 4.3. Let ${G}_{1}={K}_{n}$ and ${G}_{2}={K}_{r}$ . We get,

$\begin{array}{l}{M}_{1}\left({G}_{1}\right)=n{\left(n-1\right)}^{2},\text{ }\text{ }\\ {m}_{1}=\frac{n\left(n-1\right)}{2},\text{ }\text{ }\\ {M}_{1}\left({G}_{2}\right)=r{\left(r-1\right)}^{2},\text{ }\text{ }\\ {\stackrel{¯}{M}}_{1}\left({G}_{2}\right)=0,\end{array}$

${m}_{2}=\frac{r\left(r-1\right)}{2},\text{ }\text{ }{\stackrel{¯}{M}}_{1}\left({G}_{1}\right)=0,\text{ }\text{ }{n}_{1}=n,\text{ }\text{ }{n}_{2}=r,\text{ }\text{ }{\stackrel{¯}{m}}_{1}=0,\text{ }\text{ }{\stackrel{¯}{m}}_{2}=0.$

$\therefore$ In Theorem 4.1, put ${G}_{1}={K}_{n},\text{ }\text{ }{G}_{2}={K}_{r},$ we get

$D{D}^{*}\left[{K}_{n}+{K}_{r}\right]\le {\left[2\left(n+r-1\right)\right]}^{\frac{\left(n+r\right)\left(n+r-1\right)}{2}}$ (4)

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

5. The Multiplicative Degree Distance of Disjunction of Graph

Theorem 5.1. Let ${G}_{i},i=1,2$ , be a $\left({n}_{i},{m}_{i}\right)$ -graph and let ${\stackrel{¯}{m}}_{i}=e\left({\stackrel{¯}{G}}_{i}\right)$ . Then

$\begin{array}{l}{\left[D{D}^{*}\left({G}_{1}\vee {G}_{2}\right)\right]}^{2}\\ \le \left[\frac{1}{{n}_{1}{n}_{2}\left({n}_{1}{n}_{2}-1\right)}\left\{2{m}_{2}{n}_{2}\left(2{\stackrel{¯}{M}}_{1}\left({G}_{1}\right)+4{m}_{1}\right)+2{n}_{1}\left(2{\stackrel{¯}{m}}_{1}+{n}_{1}\right){M}_{1}\left({G}_{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-2{M}_{1}\left({G}_{2}\right)\left(2{\stackrel{¯}{m}}_{1}\left({n}_{1}-1\right)+2{m}_{1}-{M}_{1}\left({\stackrel{¯}{G}}_{1}\right)\right)+2{n}_{2}{M}_{1}\left({G}_{1}\right)\left(2{\stackrel{¯}{m}}_{2}+{n}_{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+2{m}_{1}{n}_{1}\left(2{\stackrel{¯}{M}}_{1}\left({G}_{2}\right)+4{m}_{2}\right)-2{M}_{1}\left({G}_{1}\right)\left(2\left({n}_{2}-1\right){\stackrel{¯}{m}}_{2}+2{m}_{2}-{M}_{1}\left({\stackrel{¯}{G}}_{2}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+4{n}_{2}{M}_{1}\left({G}_{1}\right){m}_{2}+4{n}_{1}{m}_{1}{M}_{1}\left({G}_{2}\right)-2{M}_{1}\left({G}_{1}\right){M}_{1}\left({G}_{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+2{n}_{2}\left(2{\stackrel{¯}{M}}_{1}\left({G}_{1}\right)+4{m}_{1}\right)\left(2{\stackrel{¯}{m}}_{2}+2{n}_{2}\right)+2{n}_{1}\left(2{\stackrel{¯}{M}}_{1}\left({G}_{2}\right)+4{m}_{2}\right)\left(2{\stackrel{¯}{m}}_{1}+{n}_{1}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-4\left(2{\stackrel{¯}{m}}_{1}\left({n}_{1}-1\right)+2{m}_{1}-{M}_{1}\left({\stackrel{¯}{G}}_{1}\right)\right)\left(2{\stackrel{¯}{m}}_{2}\left({n}_{2}-1\right)+2{m}_{2}-{M}_{1}\left({\stackrel{¯}{G}}_{2}\right)\right)\\ \text{\hspace{0.17em}}{\begin{array}{c}\\ \end{array}-8{m}_{1}{n}_{2}^{2}-4{n}_{1}^{2}{m}_{2}+16{m}_{1}{m}_{2}\right\}\right]}^{{n}_{1}{n}_{2}\left({n}_{1}{n}_{2}-1\right)}\end{array}$

Proof: ${\left[D{D}^{*}\left({G}_{1}\vee {G}_{2}\right)\right]}^{2}\le {\left[\frac{1}{{n}_{1}{n}_{2}\left({n}_{1}{n}_{2}-1\right)}\left({A}_{3}+{A}_{1}+{A}_{2}+{A}_{4}\right)\right]}^{{n}_{1}{n}_{2}\left({n}_{1}{n}_{2}-1\right)}$

where ${A}_{3},{A}_{1},{A}_{2},{A}_{4}$ are terms of the above sums taken in order.

Next we calculate ${A}_{1},{A}_{2},{A}_{3}$ and ${A}_{4}$ separately one by one. Now,

$\begin{array}{c}{A}_{1}=\underset{xy\notin {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\in {G}_{2}}{\sum }{d}_{\left({G}_{1}\vee {G}_{2}\right)}\left(\left(x,u\right),\left(y,v\right)\right)\left[{d}_{\left({G}_{1}\vee {G}_{2}\right)}\left(x,u\right)+{d}_{\left({G}_{1}\vee {G}_{2}\right)}\left(y,v\right)\right]\\ =\underset{xy\notin {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\in {G}_{2}}{\sum }1\cdot \left[{n}_{2}{d}_{{G}_{1}}\left(x\right)+{n}_{1}{d}_{{G}_{2}}\left(u\right)-{d}_{{G}_{1}}\left(x\right){d}_{{G}_{2}}\left(u\right)+{n}_{2}{d}_{{G}_{1}}\left(y\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{n}_{1}{d}_{{G}_{2}}\left(v\right)-{d}_{{G}_{1}}\left(y\right){d}_{{G}_{2}}\left(v\right)\right]\text{ }\text{ }\text{ }\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{Lemma}\text{\hspace{0.17em}}\text{2}\text{.1}\text{ }\\ ={n}_{2}\underset{xy\notin {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\in {G}_{2}}{\sum }\left[{d}_{{G}_{1}}\left(x\right)+{d}_{{G}_{1}}\left(y\right)\right]+{n}_{1}\underset{xy\notin {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\in {G}_{2}}{\sum }\left[{d}_{{G}_{2}}\left(u\right)+{d}_{{G}_{2}}\left(v\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\underset{xy\notin {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\in {G}_{2}}{\sum }{d}_{{G}_{1}}\left(x\right){d}_{{G}_{2}}\left(u\right)-\underset{xy\notin {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\in {G}_{2}}{\sum }{d}_{{G}_{1}}\left(y\right){d}_{{G}_{2}}\left(v\right)\\ ={n}_{2}\left(\underset{xy\notin {G}_{1}}{\sum }\left[{d}_{{G}_{1}}\left(x\right)+{d}_{{G}_{1}}\left(y\right)\right]\right)\left(\underset{uv\in {G}_{2}}{\sum }1\right)+{n}_{1}\left(\underset{xy\notin {G}_{1}}{\sum }1\right)\left(\underset{uv\in {G}_{2}}{\sum }\left[{d}_{{G}_{2}}\left(u\right)+{d}_{{G}_{2}}\left(v\right)\right]\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\left(\underset{xy\notin {G}_{1}}{\sum }{d}_{{G}_{1}}\left(x\right)\right)\left(\underset{uv\in {G}_{2}}{\sum }{d}_{{G}_{2}}\left(u\right)\right)-\left(\underset{xy\notin {G}_{1}}{\sum }{d}_{{G}_{1}}\left(y\right)\right)\left(\underset{uv\in {G}_{2}}{\sum }{d}_{{G}_{2}}\left(v\right)\right)\\ =2{m}_{2}{n}_{2}\left(2{\stackrel{¯}{M}}_{1}\left({G}_{1}\right)+4{m}_{1}\right)+2{n}_{1}\left(2{\stackrel{¯}{m}}_{1}+{n}_{1}\right){M}_{1}\left({G}_{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2{M}_{1}\left({G}_{2}\right)\left(2{\stackrel{¯}{m}}_{1}\left({n}_{1}-1\right)+2{m}_{1}-{M}_{1}\left({\stackrel{¯}{G}}_{1}\right)\right)\end{array}$

$\begin{array}{c}{A}_{2}=\underset{xy\in {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2}}{\sum }{d}_{\left({G}_{1}\vee {G}_{2}\right)}\left(\left(x,u\right),\left(y,v\right)\right)\left[{d}_{\left({G}_{1}\vee {G}_{2}\right)}\left(x,u\right)+{d}_{\left({G}_{1}\vee {G}_{2}\right)}\left(y,v\right)\right]\\ =\underset{xy\in {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2}}{\sum }\left[{n}_{2}{d}_{{G}_{1}}\left(x\right)+{n}_{1}{d}_{{G}_{2}}\left(u\right)-{d}_{{G}_{1}}\left(x\right){d}_{{G}_{2}}\left(u\right)+{n}_{2}{d}_{{G}_{1}}\left(y\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{n}_{1}{d}_{{G}_{2}}\left(v\right)-{d}_{{G}_{1}}\left(y\right){d}_{{G}_{2}}\left(v\right)\right]\text{ }\text{ }\text{ }\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{Lemma}\text{\hspace{0.17em}}\text{2}\text{.1}\\ ={n}_{2}\underset{xy\in {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2}}{\sum }\left[{d}_{{G}_{1}}\left(x\right)+{d}_{{G}_{2}}\left(y\right)\right]+{n}_{1}\underset{xy\in {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2}}{\sum }\left[{d}_{{G}_{2}}\left(u\right)+{d}_{{G}_{2}}\left(v\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\underset{xy\in {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2}}{\sum }{d}_{{G}_{1}}\left(x\right){d}_{{G}_{2}}\left(u\right)-\underset{xy\in {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2}}{\sum }{d}_{{G}_{1}}\left(y\right){d}_{{G}_{2}}\left(v\right)\\ ={n}_{2}\left(\underset{xy\in {G}_{1}}{\sum }\left[{d}_{{G}_{1}}\left(x\right)+{d}_{{G}_{2}}\left(y\right)\right]\right)\left(\underset{uv\notin {G}_{2}}{\sum }1\right)+{n}_{1}\left(\underset{xy\in {G}_{1}}{\sum }1\right)\left(\underset{uv\notin {G}_{2}}{\sum }\left[{d}_{{G}_{2}}\left(u\right)+{d}_{{G}_{2}}\left(v\right)\right]\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\left(\underset{xy\in {G}_{1}}{\sum }{d}_{{G}_{1}}\left(x\right)\right)\left(\underset{uv\notin {G}_{2}}{\sum }{d}_{{G}_{2}}\left(u\right)\right)-\left(\underset{xy\in {G}_{1}}{\sum }{d}_{{G}_{1}}\left(y\right)\right)\left(\underset{uv\notin {G}_{2}}{\sum }{d}_{{G}_{2}}\left(v\right)\right)\\ =2{n}_{2}{M}_{1}\left({G}_{1}\right)\left(2{\stackrel{¯}{m}}_{2}+{n}_{2}\right)+2{n}_{1}{m}_{1}\left(2{\stackrel{¯}{M}}_{1}\left({G}_{2}\right)+4{m}_{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2{M}_{1}\left({G}_{1}\right)\left[2\left({n}_{2}-1\right){\stackrel{¯}{m}}_{2}+2{m}_{2}-{M}_{1}\left({\stackrel{¯}{G}}_{2}\right)\right]\end{array}$

$\begin{array}{c}{A}_{3}=\underset{xy\in {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\in {G}_{2}}{\sum }{d}_{\left({G}_{1}\vee {G}_{2}\right)}\left(\left(x,u\right),\left(y,v\right)\right)\left[{d}_{\left({G}_{1}\vee {G}_{2}\right)}\left(x,u\right)+{d}_{\left({G}_{1}\vee {G}_{2}\right)}\left(y,v\right)\right]\\ =\underset{xy\in {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\in {G}_{2}}{\sum }\left[{n}_{2}{d}_{{G}_{1}}\left(x\right)+{n}_{1}{d}_{{G}_{2}}\left(u\right)-{d}_{{G}_{1}}\left(x\right){d}_{{G}_{2}}\left(u\right)+{n}_{2}{d}_{{G}_{1}}\left(y\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{n}_{1}{d}_{{G}_{2}}\left(v\right)-{d}_{{G}_{1}}\left(y\right){d}_{{G}_{2}}\left(v\right)\right]\text{ }\text{ }\text{ }\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{Lemma}\text{\hspace{0.17em}}\text{2}\text{.1}\text{ }\\ ={n}_{2}\underset{xy\in {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\in {G}_{2}}{\sum }\left[{d}_{{G}_{1}}\left(x\right)+{d}_{{G}_{2}}\left(y\right)\right]+{n}_{1}\underset{xy\in {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\in {G}_{2}}{\sum }\left[{d}_{{G}_{2}}\left(u\right)+{d}_{{G}_{2}}\left(v\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\underset{xy\in {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\in {G}_{2}}{\sum }{d}_{{G}_{1}}\left(x\right){d}_{{G}_{2}}\left(u\right)-\underset{xy\in {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\in {G}_{2}}{\sum }{d}_{{G}_{1}}\left(y\right){d}_{{G}_{2}}\left(v\right)\\ ={n}_{2}\left(\underset{xy\in {G}_{1}}{\sum }\left[{d}_{{G}_{1}}\left(x\right)+{d}_{{G}_{2}}\left(y\right)\right]\right)\left(\underset{uv\in {G}_{2}}{\sum }1\right)+{n}_{1}\left(\underset{xy\in {G}_{1}}{\sum }1\right)\left(\underset{uv\in {G}_{2}}{\sum }\left[{d}_{{G}_{2}}\left(u\right)+{d}_{{G}_{2}}\left(v\right)\right]\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\left(\underset{xy\in {G}_{1}}{\sum }{d}_{{G}_{1}}\left(x\right)\right)\left(\underset{uv\in {G}_{2}}{\sum }{d}_{{G}_{2}}\left(u\right)\right)-\left(\underset{xy\in {G}_{1}}{\sum }{d}_{{G}_{1}}\left(y\right)\right)\left(\underset{uv\in {G}_{2}}{\sum }{d}_{{G}_{2}}\left(v\right)\right)\\ =4{n}_{2}{m}_{2}{M}_{1}\left({G}_{1}\right)+4{n}_{1}{m}_{1}{M}_{1}\left({G}_{2}\right)-2{M}_{1}\left({G}_{1}\right){M}_{1}\left( G 2 \right)\end{array}$

$\begin{array}{c}{A}_{4}=\underset{xy\notin {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2}}{\sum }{d}_{\left({G}_{1}\vee {G}_{2}\right)}\left(\left(x,u\right),\left(y,v\right)\right)\left[{d}_{\left({G}_{1}\vee {G}_{2}\right)}\left(x,u\right)+{d}_{\left({G}_{1}\vee {G}_{2}\right)}\left(y,v\right)\right]\\ =2\underset{xy\notin {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2}}{\sum }\left[{d}_{\left({G}_{1}\vee {G}_{2}\right)}\left(x,u\right)+{d}_{\left({G}_{1}\vee {G}_{2}\right)}\left(y,v\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2\underset{xy\notin {G}_{1},x=y}{\sum }\text{ }\text{ }\text{ }\text{ }\underset{uv\notin {G}_{2},u=v}{\sum }\left[{d}_{\left({G}_{1}\vee {G}_{2}\right)}\left(x,u\right)+{d}_{\left({G}_{1}\vee {G}_{2}\right)}\left(y,v\right)\right]\end{array}$

${A}_{4}=2{A}_{5}-2{A}_{6},$ where ${A}_{5}$ and ${A}_{6}$ are terms of the above sums taken in order.

Now,

$\begin{array}{c}{A}_{5}=\underset{xy\notin {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2}}{\sum }\left[{d}_{\left({G}_{1}\vee {G}_{2}\right)}\left(x,u\right)+{d}_{\left({G}_{1}\vee {G}_{2}\right)}\left(y,v\right)\right]\\ =\underset{xy\notin {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2}}{\sum }\left[{n}_{2}{d}_{{G}_{1}}\left(x\right)+{n}_{1}{d}_{{G}_{2}}\left(u\right)-{d}_{{G}_{1}}\left(x\right){d}_{{G}_{2}}\left(u\right)+{n}_{2}{d}_{{G}_{1}}\left(y\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{n}_{1}{d}_{{G}_{2}}\left(v\right)-{d}_{{G}_{1}}\left(y\right){d}_{{G}_{2}}\left(v\right)\right]\text{ }\text{\hspace{0.17em}}\text{ }\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{Lemma}\text{\hspace{0.17em}}\text{2}\text{.1}\text{ }\\ ={n}_{2}\underset{xy\notin {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2}}{\sum }\left[{d}_{{G}_{1}}\left(x\right)+{d}_{{G}_{1}}\left(y\right)\right]+{n}_{1}\underset{xy\notin {G}_{1}}{\sum }\underset{uv\notin {G}_{2}}{\sum }\left[{d}_{{G}_{2}}\left(u\right)+{d}_{{G}_{2}}\left(v\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\underset{xy\notin {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2}}{\sum }{d}_{{G}_{1}}\left(x\right){d}_{{G}_{2}}\left(u\right)-\underset{xy\notin {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2}}{\sum }{d}_{{G}_{1}}\left(y\right){d}_{{G}_{2}}\left(v\right)\\ ={n}_{2}\left(\underset{xy\notin {G}_{1}}{\sum }\left[{d}_{{G}_{1}}\left(x\right)+{d}_{{G}_{2}}\left(y\right)\right]\right)\left(\underset{uv\notin {G}_{2}}{\sum }1\right)+{n}_{1}\left(\underset{xy\notin {G}_{1}}{\sum }1\right)\left(\underset{uv\notin {G}_{2}}{\sum }\left[{d}_{{G}_{2}}\left(u\right)+{d}_{{G}_{2}}\left(v\right)\right]\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\left(\underset{xy\notin {G}_{1}}{\sum }{d}_{{G}_{1}}\left(x\right)\right)\left(\underset{uv\notin {G}_{2}}{\sum }{d}_{{G}_{2}}\left(u\right)\right)-\left(\underset{xy\notin {G}_{1}}{\sum }{d}_{{G}_{1}}\left(y\right)\right)\left(\underset{uv\notin {G}_{2}}{\sum }{d}_{{G}_{2}}\left(v\right)\right)\\ ={n}_{2}\left(2{\stackrel{¯}{M}}_{1}\left({G}_{1}\right)+4{m}_{1}\right)\left(2{\stackrel{¯}{m}}_{2}+{n}_{2}\right)+{n}_{1}\left(2{\stackrel{¯}{M}}_{1}\left({G}_{2}\right)+4{m}_{2}\right)\left(2{\stackrel{¯}{m}}_{1}+{n}_{1}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2\left(2{\stackrel{¯}{m}}_{1}\left({n}_{1}-1\right)+2{m}_{1}-{M}_{1}\left({\stackrel{¯}{G}}_{1}\right)\right)\left(2{\stackrel{¯}{m}}_{2}\left({n}_{2}-1\right)+2{m}_{2}-{M}_{1}\left({\stackrel{¯}{G}}_{2}\right)\right)\end{array}$

$\begin{array}{c}{A}_{6}=\underset{xy\notin {G}_{1},x=y}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2},u=v}{\sum }\left[{d}_{\left({G}_{1}\vee {G}_{2}\right)}\left(x,u\right)+{d}_{\left({G}_{1}\vee {G}_{2}\right)}\left(y,v\right)\right]\\ =\underset{xy\notin {G}_{1},u=v}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2},u=v}{\sum }\left[{n}_{2}{d}_{{G}_{1}}\left(x\right)+{n}_{1}{d}_{{G}_{2}}\left(u\right)-{d}_{{G}_{1}}\left(x\right){d}_{{G}_{2}}\left(u\right)+{n}_{2}{d}_{{G}_{1}}\left(y\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{n}_{1}{d}_{{G}_{2}}\left(v\right)-{d}_{{G}_{1}}\left(y\right){d}_{{G}_{2}}\left(v\right)\right]\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }\text{by}\text{\hspace{0.17em}}\text{Lemma}\text{\hspace{0.17em}}\text{2}\text{.1}\text{ }\\ ={n}_{2}\underset{xy\notin {G}_{1},x=y}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2},u=v}{\sum }\left[{d}_{{G}_{1}}\left(x\right)+{d}_{{G}_{1}}\left(y\right)\right]+{n}_{1}\underset{xy\notin {G}_{1},x=y}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2},u=v}{\sum }\left[{d}_{{G}_{2}}\left(u\right)+{d}_{{G}_{2}}\left(v\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\underset{xy\notin {G}_{1},x=y}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2},u=v}{\sum }{d}_{{G}_{1}}\left(x\right){d}_{{G}_{2}}\left(u\right)-\underset{xy\notin {G}_{1},x=y}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2},u=v}{\sum }{d}_{{G}_{1}}\left(y\right){d}_{{G}_{2}}\left(v\right)\\ ={n}_{2}\left(\underset{xy\notin {G}_{1},x=y}{\sum }\left[{d}_{{G}_{1}}\left(x\right)+{d}_{{G}_{2}}\left(y\right)\right]\right)\left(\underset{uv\notin {G}_{2},u=v}{\sum }1\right)+{n}_{1}\left(\underset{xy\notin {G}_{1},x=y}{\sum }1\right)\left(\underset{uv\notin {G}_{2},u=v}{\sum }\left[{d}_{{G}_{2}}\left(u\right)+{d}_{{G}_{2}}\left(v\right)\right]\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\left(\underset{xy\notin {G}_{1},x=y}{\sum }{d}_{{G}_{1}}\left(x\right)\right)\left(\underset{uv\notin {G}_{2},u=v}{\sum }{d}_{{G}_{2}}\left(u\right)\right)-\left(\underset{xy\notin {G}_{1},x=y}{\sum }{d}_{{G}_{1}}\left(y\right)\right)\left(\underset{uv\notin {G}_{2},u=v}{\sum }{d}_{{G}_{2}}\left(v\right)\right)\\ =4{m}_{1}{n}_{2}^{2}+4{m}_{2}{n}_{1}^{2}-8{m}_{1}{m}_{2}\end{array}$

$\begin{array}{l}{\left[D{D}^{*}\left({G}_{1}\vee {G}_{2}\right)\right]}^{2}\\ \le \left[\frac{1}{{n}_{1}{n}_{2}\left({n}_{1}{n}_{2}-1\right)}\left\{2{m}_{2}{n}_{2}\left(2{\stackrel{¯}{M}}_{1}\left({G}_{1}\right)+4{m}_{1}\right)+2{n}_{1}\left(2{\stackrel{¯}{m}}_{1}+{n}_{1}\right){M}_{1}\left({G}_{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-2{M}_{1}\left({G}_{2}\right)\left(2{\stackrel{¯}{m}}_{1}\left({n}_{1}-1\right)+2{m}_{1}-{M}_{1}\left({\stackrel{¯}{G}}_{1}\right)\right)+2{n}_{2}{M}_{1}\left({G}_{1}\right)\left(2{\stackrel{¯}{m}}_{2}+{n}_{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+2{m}_{1}{n}_{1}\left(2{\stackrel{¯}{M}}_{1}\left({G}_{2}\right)+4{m}_{2}\right)-2{M}_{1}\left({G}_{1}\right)\left(2\left({n}_{2}-1\right){\stackrel{¯}{m}}_{2}+2{m}_{2}-{M}_{1}\left({\stackrel{¯}{G}}_{2}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+4{n}_{2}{M}_{1}\left({G}_{1}\right){m}_{2}+4{n}_{1}{m}_{1}{M}_{1}\left({G}_{2}\right)-2{M}_{1}\left({G}_{1}\right){M}_{1}\left({G}_{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+2{n}_{2}\left(2{\stackrel{¯}{M}}_{1}\left({G}_{1}\right)+4{m}_{1}\right)\left(2{\stackrel{¯}{m}}_{2}+2{n}_{2}\right)+2{n}_{1}\left(2{\stackrel{¯}{M}}_{1}\left({G}_{2}\right)+4{m}_{2}\right)\left(2{\stackrel{¯}{m}}_{1}+{n}_{1}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-4\left(2{\stackrel{¯}{m}}_{1}\left({n}_{1}-1\right)+2{m}_{1}-{M}_{1}\left({\stackrel{¯}{G}}_{1}\right)\right)\left(2{\stackrel{¯}{m}}_{2}\left({n}_{2}-1\right)+2{m}_{2}-{M}_{1}\left({\stackrel{¯}{G}}_{2}\right)\right)\\ \text{\hspace{0.17em}}{\begin{array}{c}\\ \end{array}-8{m}_{1}{n}_{2}^{2}-4{n}_{1}^{2}{m}_{2}+16{m}_{1}{m}_{2}\right\}\right]}^{{n}_{1}{n}_{2}\left({n}_{1}{n}_{2}-1\right)}\end{array}$

Lemma 5.2.

$D{D}^{*}\left[{K}_{m}\vee {K}_{n}\right]={\left(2mn-2\right)}^{\frac{mn\left(mn-1\right)}{2}}$

Proof: Clearly the graph ${K}_{m}\vee {K}_{n}$ is the complete graph ${K}_{mn}$ .

$D{D}^{*}\left({K}_{m}\vee {K}_{n}\right)=D{D}^{*}\left({K}_{mn}\right)={\left(2mn-2\right)}^{\frac{mn\left(mn-1\right)}{2}}$ (5)

Remark 5.3. Let ${G}_{1}={K}_{m}$ and ${G}_{2}={K}_{n}$ . We get

${n}_{1}=m,\text{ }\text{ }{n}_{2}=n,\text{ }\text{ }{m}_{1}=\frac{m\left(m-1\right)}{2},\text{ }\text{ }{m}_{2}=\frac{n\left(n-1\right)}{2},\text{\hspace{0.17em}}\text{ }{\stackrel{¯}{m}}_{1}=0,\text{ }\text{ }{\stackrel{¯}{m}}_{2}=0$

${M}_{1}\left({G}_{1}\right)={M}_{1}\left({K}_{m}\right)=m{\left(m-1\right)}^{2},\text{ }\text{ }{M}_{1}\left({G}_{2}\right)={M}_{1}\left({K}_{n}\right)=n{\left(n-1\right)}^{2}$

${M}_{1}\left({\stackrel{¯}{G}}_{1}\right)={M}_{1}\left({\stackrel{¯}{K}}_{m}\right)=0,\text{ }\text{ }{M}_{1}\left({\stackrel{¯}{G}}_{2}\right)={M}_{1}\left({\stackrel{¯}{K}}_{n}\right)=0,\text{ }\text{ }{\stackrel{¯}{M}}_{1}\left({G}_{1}\right)={\stackrel{¯}{M}}_{1}\left({K}_{m}\right)=0$

$\therefore$ In Theorem 5.1, put ${G}_{1}={K}_{m}$ and ${G}_{2}={K}_{n}$ , we get

$D{D}^{*}\left[{K}_{m}\vee {K}_{n}\right]\le {\left(2mn-2\right)}^{\frac{mn\left(mn-1\right)}{2}}$ (6)

From (5) and (6) our bound is tight.

6. The Multiplicative Degree Distance of Symmetric difference of Graph

Theorem 6.1.

$\begin{array}{l}{\left[D{D}^{*}\left({G}_{1}\oplus {G}_{2}\right)\right]}^{2}\\ \le \left\{\frac{1}{{n}_{1}{n}_{2}\left({n}_{1}{n}_{2}-1\right)}\left[2{n}_{2}{m}_{2}\left(2{\stackrel{¯}{M}}_{1}\left({G}_{1}\right)+4{m}_{1}\right)+2{n}_{1}{M}_{1}\left({G}_{2}\right)\left(2{\stackrel{¯}{m}}_{1}+{n}_{1}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-4{M}_{1}\left({G}_{2}\right)\left(2{\stackrel{¯}{m}}_{1}\left({n}_{1}-1\right)+2{m}_{1}-{M}_{1}\left({\stackrel{¯}{G}}_{1}\right)\right)+2{n}_{2}{M}_{1}\left({G}_{1}\right)\left(2{\stackrel{¯}{m}}_{2}+{n}_{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+2{n}_{1}{m}_{1}\left(2{\stackrel{¯}{M}}_{1}\left({G}_{2}\right)+4{m}_{2}\right)-4{M}_{1}\left({G}_{1}\right)\left(2\left({n}_{2}-1\right){\stackrel{¯}{m}}_{2}+2{m}_{2}-{M}_{1}\left({\stackrel{¯}{G}}_{2}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+4{n}_{2}{M}_{1}\left({G}_{1}\right){m}_{2}+4{n}_{1}{m}_{1}{M}_{1}\left({G}_{2}\right)-4{M}_{1}\left({G}_{1}\right){M}_{1}\left({G}_{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\left[{n}_{2}\left(2{\stackrel{¯}{M}}_{1}\left({G}_{1}\right)+4{m}_{1}\right)\left(2{m}_{2}+{n}_{2}\right)+{n}_{1}\left(2{\stackrel{¯}{M}}_{1}\left({G}_{2}\right)+4{m}_{2}\right)\left(2{m}_{1}+{n}_{1}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-4\left(2{\stackrel{¯}{m}}_{1}\left({n}_{1}-1\right)+2{m}_{1}-{M}_{1}\left({\stackrel{¯}{G}}_{1}\right)\right)\left(2{\stackrel{¯}{m}}_{2}\left({n}_{2}-1\right)+2{m}_{2}-{M}_{1}\left({\stackrel{¯}{G}}_{2}\right)\right)\right]\\ \text{\hspace{0.17em}}{\begin{array}{c}\\ \end{array}-2\left(4{n}_{2}^{2}{m}_{1}+4{n}_{1}^{2}{m}_{2}-16{m}_{1}{m}_{2}\right)\right]\right\}}^{{n}_{1}{n}_{2}\left({n}_{1}{n}_{2}-1\right)}\end{array}$

Proof: ${\left[D{D}^{*}\left({G}_{1}\oplus {G}_{2}\right)\right]}^{2}\le {\left[\frac{1}{{n}_{1}{n}_{2}\left({n}_{1}{n}_{2}-1\right)}\left({C}_{3}+{C}_{1}+{C}_{2}+{C}_{4}\right)\right]}^{{n}_{1}{n}_{2}\left({n}_{1}{n}_{2}-1\right)}$

where ${C}_{3},{C}_{1},{C}_{2},{C}_{4}$ are terms of the above sums taken in order.

Next we calculate ${C}_{1},{C}_{2},{C}_{3}$ and ${C}_{4}$ separately.

$\begin{array}{c}{C}_{1}=\underset{xy\notin {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\in {G}_{2}}{\sum }{d}_{\left({G}_{1}\oplus {G}_{2}\right)}\left(\left(x,u\right),\left(y,v\right)\right)\left[{d}_{\left({G}_{1}\oplus {G}_{2}\right)}\left(x,u\right)+{d}_{\left({G}_{1}\oplus {G}_{2}\right)}\left(y,v\right)\right]\\ =\underset{xy\notin {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\in {G}_{2}}{\sum }1\cdot \left[{n}_{2}{d}_{{G}_{1}}\left(x\right)+{n}_{1}{d}_{{G}_{2}}\left(u\right)-2{d}_{{G}_{1}}\left(x\right){d}_{{G}_{2}}\left(u\right)+{n}_{2}{d}_{{G}_{1}}\left(y\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{n}_{1}{d}_{{G}_{2}}\left(v\right)-2{d}_{{G}_{1}}\left(y\right){d}_{{G}_{2}}\left(v\right)\right]\text{ }\text{ }\text{ }\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{Lemma}\text{\hspace{0.17em}}\text{2}\text{.1}\text{ }\\ ={n}_{2}\underset{xy\notin {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\in {G}_{2}}{\sum }\left[{d}_{{G}_{1}}\left(x\right)+{d}_{{G}_{1}}\left(y\right)\right]+{n}_{1}\underset{xy\notin {G}_{1}}{\sum }\underset{uv\in {G}_{2}}{\sum }\left[{d}_{{G}_{2}}\left(u\right)+{d}_{{G}_{2}}\left(v\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2\underset{xy\notin {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\in {G}_{2}}{\sum }{d}_{{G}_{1}}\left(x\right){d}_{{G}_{2}}\left(u\right)-2\underset{xy\notin {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\in {G}_{2}}{\sum }{d}_{{G}_{1}}\left(y\right){d}_{{G}_{2}}\left(v\right)\\ ={n}_{2}\left(\underset{xy\notin {G}_{1}}{\sum }\left[{d}_{{G}_{1}}\left(x\right)+{d}_{{G}_{1}}\left(y\right)\right]\right)\left(\underset{uv\in {G}_{2}}{\sum }1\right)+{n}_{1}\left(\underset{xy\notin {G}_{1}}{\sum }1\right)\left(\underset{uv\in {G}_{2}}{\sum }\left[{d}_{{G}_{2}}\left(u\right)+{d}_{{G}_{2}}\left(v\right)\right]\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2\left(\underset{xy\notin {G}_{1}}{\sum }{d}_{{G}_{1}}\left(x\right)\right)\left(\underset{uv\in {G}_{2}}{\sum }{d}_{{G}_{2}}\left(u\right)\right)-2\left(\underset{xy\notin {G}_{1}}{\sum }{d}_{{G}_{1}}\left(y\right)\right)\left(\underset{uv\in {G}_{2}}{\sum }{d}_{{G}_{2}}\left(v\right)\right)\\ =2{n}_{2}{m}_{2}\left(2{\stackrel{¯}{M}}_{1}\left({G}_{1}\right)+4{m}_{1}\right)+2{n}_{1}{M}_{1}\left({G}_{2}\right)\left(2{\stackrel{¯}{m}}_{1}+{n}_{1}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-4{M}_{1}\left({G}_{2}\right)\left(2{\stackrel{¯}{m}}_{1}\left({n}_{1}-1\right)+2{m}_{1}-{M}_{1}\left({\stackrel{¯}{G}}_{1}\right)\right)\end{array}$

$\begin{array}{c}{C}_{2}=\underset{xy\in {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2}}{\sum }{d}_{\left({G}_{1}\oplus {G}_{2}\right)}\left(\left(x,u\right),\left(y,v\right)\right)\left[{d}_{\left({G}_{1}\oplus {G}_{2}\right)}\left(x,u\right)+{d}_{\left({G}_{1}\oplus {G}_{2}\right)}\left(y,v\right)\right]\\ =\underset{xy\in {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2}}{\sum }\left[{n}_{2}{d}_{{G}_{1}}\left(x\right)+{n}_{1}{d}_{{G}_{2}}\left(u\right)-2{d}_{{G}_{1}}\left(x\right){d}_{{G}_{2}}\left(u\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{n}_{2}{d}_{{G}_{1}}\left(y\right)+{n}_{1}{d}_{{G}_{2}}\left(v\right)-2{d}_{{G}_{1}}\left(y\right){d}_{{G}_{2}}\left(v\right)\right]\text{ }\text{ }\text{ }\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{Lemma}\text{\hspace{0.17em}}\text{2}\text{.1}\text{ }\\ ={n}_{2}\underset{xy\in {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2}}{\sum }\left[{d}_{{G}_{1}}\left(x\right)+{d}_{{G}_{2}}\left(y\right)\right]+{n}_{1}\underset{xy\in {G}_{1}}{\sum }\underset{uv\notin {G}_{2}}{\sum }\left[{d}_{{G}_{2}}\left(u\right)+{d}_{{G}_{2}}\left(v\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2\underset{xy\in {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2}}{\sum }{d}_{{G}_{1}}\left(x\right){d}_{{G}_{2}}\left(u\right)-2\underset{xy\in {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2}}{\sum }{d}_{{G}_{1}}\left(y\right){d}_{{G}_{2}}\left(v\right)\\ ={n}_{2}\left(\underset{xy\in {G}_{1}}{\sum }\left[{d}_{{G}_{1}}\left(x\right)+{d}_{{G}_{2}}\left(y\right)\right]\right)\left(\underset{uv\notin {G}_{2}}{\sum }1\right)+{n}_{1}\left(\underset{xy\in {G}_{1}}{\sum }1\right)\left(\underset{uv\notin {G}_{2}}{\sum }\left[{d}_{{G}_{2}}\left(u\right)+{d}_{{G}_{2}}\left(v\right)\right]\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2\left(\underset{xy\in {G}_{1}}{\sum }{d}_{{G}_{1}}\left(x\right)\right)\left(\underset{uv\notin {G}_{2}}{\sum }{d}_{{G}_{2}}\left(u\right)\right)-2\left(\underset{xy\in {G}_{1}}{\sum }{d}_{{G}_{1}}\left(y\right)\right)\left(\underset{uv\notin {G}_{2}}{\sum }{d}_{{G}_{2}}\left(v\right)\right)\\ =2{n}_{2}{M}_{1}\left({G}_{1}\right)\left({\stackrel{¯}{m}}_{2}+{n}_{2}\right)+2{n}_{1}{m}_{1}\left(2{\stackrel{¯}{M}}_{1}\left({G}_{2}\right)+4{m}_{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-4{M}_{1}\left({G}_{1}\right)\left(2\left({n}_{2}-1\right){\stackrel{¯}{m}}_{2}+2{m}_{2}-{M}_{1}\left({\stackrel{¯}{G}}_{2}\right)\right)\end{array}$

$\begin{array}{c}{C}_{3}=\underset{xy\in {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\in {G}_{2}}{\sum }{d}_{\left({G}_{1}\oplus {G}_{2}\right)}\left(\left(x,u\right),\left(y,v\right)\right)\left[{d}_{\left({G}_{1}\oplus {G}_{2}\right)}\left(x,u\right)+{d}_{\left({G}_{1}\oplus {G}_{2}\right)}\left(y,v\right)\right]\\ =\underset{xy\in {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\in {G}_{2}}{\sum }\left[{n}_{2}{d}_{{G}_{1}}\left(x\right)+{n}_{1}{d}_{{G}_{2}}\left(u\right)-2{d}_{{G}_{1}}\left(x\right){d}_{{G}_{2}}\left(u\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{n}_{2}{d}_{{G}_{1}}\left(y\right)+{n}_{1}{d}_{{G}_{2}}\left(v\right)-2{d}_{{G}_{1}}\left(y\right){d}_{{G}_{2}}\left(v\right)\right]\text{ }\text{ }\text{ }\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{Lemma}\text{\hspace{0.17em}}\text{2}\text{.1}\text{ }\\ ={n}_{2}\underset{xy\in {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\in {G}_{2}}{\sum }\left[{d}_{{G}_{1}}\left(x\right)+{d}_{{G}_{2}}\left(y\right)\right]+{n}_{1}\underset{xy\in {G}_{1}}{\sum }\text{ }\text{ }\text{ }\underset{uv\in {G}_{2}}{\sum }\left[{d}_{{G}_{2}}\left(u\right)+{d}_{{G}_{2}}\left(v\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2\underset{xy\in {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\in {G}_{2}}{\sum }{d}_{{G}_{1}}\left(x\right){d}_{{G}_{2}}\left(u\right)-2\underset{xy\in {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\in {G}_{2}}{\sum }{d}_{{G}_{1}}\left(y\right){d}_{{G}_{2}}\left(v\right)\\ ={n}_{2}\left(\underset{xy\in {G}_{1}}{\sum }\left[{d}_{{G}_{1}}\left(x\right)+{d}_{{G}_{2}}\left(y\right)\right]\right)\left(\underset{uv\in {G}_{2}}{\sum }1\right)+{n}_{1}\left(\underset{xy\in {G}_{1}}{\sum }1\right)\left(\underset{uv\in {G}_{2}}{\sum }\left[{d}_{{G}_{2}}\left(u\right)+{d}_{{G}_{2}}\left(v\right)\right]\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2\left(\underset{xy\in {G}_{1}}{\sum }{d}_{{G}_{1}}\left(x\right)\right)\left(\underset{uv\in {G}_{2}}{\sum }{d}_{{G}_{2}}\left(u\right)\right)-2\left(\underset{xy\in {G}_{1}}{\sum }{d}_{{G}_{1}}\left(y\right)\right)\left(\underset{uv\in {G}_{2}}{\sum }{d}_{{G}_{2}}\left(v\right)\right)\\ =4{n}_{2}{M}_{1}\left({G}_{1}\right){m}_{2}+4{n}_{1}{m}_{1}{M}_{1}\left({G}_{2}\right)-4{M}_{1}\left({G}_{1}\right){M}_{1}\left( G 2 \right)\end{array}$

$\begin{array}{c}{C}_{4}=\underset{xy\notin {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2}}{\sum }{d}_{\left({G}_{1}\oplus {G}_{2}\right)}\left(\left(x,u\right),\left(y,v\right)\right)\left[{d}_{\left({G}_{1}\oplus {G}_{2}\right)}\left(x,u\right)+{d}_{\left({G}_{1}\oplus {G}_{2}\right)}\left(y,v\right)\right]\\ =2\underset{xy\notin {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2}}{\sum }\left[{d}_{\left({G}_{1}\oplus {G}_{2}\right)}\left(x,u\right)+{d}_{\left({G}_{1}\oplus {G}_{2}\right)}\left(y,v\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2\underset{xy\notin {G}_{1},x=y}{\sum }\text{ }\text{ }\text{ }\underset{uv\notin {G}_{2},u=v}{\sum }\left[{d}_{\left({G}_{1}\oplus {G}_{2}\right)}\left(x,u\right)+{d}_{\left({G}_{1}\oplus {G}_{2}\right)}\left(y,v\right)\right]\end{array}$

${C}_{4}=2{C}_{5}-2{C}_{6},$ where ${C}_{5}$ and ${C}_{6}$ denote the sums of the above terms in order.

Now,

$\begin{array}{c}{C}_{5}=\underset{xy\notin {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2}}{\sum }\left[{d}_{\left({G}_{1}\oplus {G}_{2}\right)}\left(x,u\right)+{d}_{\left({G}_{1}\oplus {G}_{2}\right)}\left(y,v\right)\right]\\ =\underset{xy\notin {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2}}{\sum }\left[{n}_{2}{d}_{{G}_{1}}\left(x\right)+{n}_{1}{d}_{{G}_{2}}\left(u\right)-2{d}_{{G}_{1}}\left(x\right){d}_{{G}_{2}}\left(u\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{n}_{2}{d}_{{G}_{1}}\left(y\right)+{n}_{1}{d}_{{G}_{2}}\left(v\right)-2{d}_{{G}_{1}}\left(y\right){d}_{{G}_{2}}\left(v\right)\right]\text{ }\text{ }\text{\hspace{0.17em}}\text{ }\text{by}\text{\hspace{0.17em}}\text{Lemma}\text{\hspace{0.17em}}\text{2}\text{.1}\text{ }\\ ={n}_{2}\underset{xy\notin {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2}}{\sum }\left[{d}_{{G}_{1}}\left(x\right)+{d}_{{G}_{2}}\left(y\right)\right]+{n}_{1}\underset{xy\notin {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2}}{\sum }\left[{d}_{{G}_{2}}\left(u\right)+{d}_{{G}_{2}}\left(v\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2\underset{xy\notin {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2}}{\sum }{d}_{{G}_{1}}\left(x\right){d}_{{G}_{2}}\left(u\right)-2\underset{xy\notin {G}_{1}}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2}}{\sum }{d}_{{G}_{1}}\left(y\right){d}_{{G}_{2}}\left(v\right)\\ ={n}_{2}\left(\underset{xy\notin {G}_{1}}{\sum }\left[{d}_{{G}_{1}}\left(x\right)+{d}_{{G}_{2}}\left(y\right)\right]\right)\left(\underset{uv\notin {G}_{2}}{\sum }1\right)+{n}_{1}\left(\underset{xy\notin {G}_{1}}{\sum }1\right)\left(\underset{uv\notin {G}_{2}}{\sum }\left[{d}_{{G}_{2}}\left(u\right)+{d}_{{G}_{2}}\left(v\right)\right]\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2\left(\underset{xy\notin {G}_{1}}{\sum }{d}_{{G}_{1}}\left(x\right)\right)\left(\underset{uv\notin {G}_{2}}{\sum }{d}_{{G}_{2}}\left(u\right)\right)-2\left(\underset{xy\notin {G}_{1}}{\sum }{d}_{{G}_{1}}\left(y\right)\right)\left(\underset{uv\notin {G}_{2}}{\sum }{d}_{{G}_{2}}\left(v\right)\right)\\ ={n}_{2}\left(2{\stackrel{¯}{M}}_{1}\left({G}_{1}\right)+4{m}_{1}\right)\left(2{m}_{2}+{n}_{2}\right)+{n}_{1}\left(2{\stackrel{¯}{M}}_{1}\left({G}_{2}\right)+4{m}_{2}\right)\left(2{m}_{1}+{n}_{1}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-4\left(\left(2{\stackrel{¯}{m}}_{1}\left({n}_{1}-1\right)+2{m}_{1}-{M}_{1}\left({\stackrel{¯}{G}}_{1}\right)\right)\left(2{\stackrel{¯}{m}}_{2}\left({n}_{2}-1\right)+2{m}_{2}-{M}_{1}\left({\stackrel{¯}{G}}_{2}\right)\right)\end{array}$

$\begin{array}{c}{C}_{6}=\underset{xy\notin {G}_{1},x=y}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2},u=v}{\sum }\left[{d}_{\left({G}_{1}\oplus {G}_{2}\right)}\left(x,u\right)+{d}_{\left({G}_{1}\oplus {G}_{2}\right)}\left(y,v\right)\right]\\ =\underset{xy\notin {G}_{1},x=y}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2},u=v}{\sum }\left[{n}_{2}{d}_{{G}_{1}}\left(x\right)+{n}_{1}{d}_{{G}_{2}}\left(u\right)-2{d}_{{G}_{1}}\left(x\right){d}_{{G}_{2}}\left(u\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{n}_{2}{d}_{{G}_{1}}\left(y\right)+{n}_{1}{d}_{{G}_{2}}\left(v\right)-2{d}_{{G}_{1}}\left(y\right){d}_{{G}_{2}}\left(v\right)\right]\text{ }\text{ }\text{ }\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{Lemma}\text{\hspace{0.17em}}\text{2}\text{.1}\text{ }\\ ={n}_{2}\underset{xy\notin {G}_{1},x=y}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2},u=v}{\sum }\left[{d}_{{G}_{1}}\left(x\right)+{d}_{{G}_{2}}\left(y\right)\right]+{n}_{1}\underset{xy\notin {G}_{1},x=y}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2},u=v}{\sum }\left[{d}_{{G}_{2}}\left(u\right)+{d}_{{G}_{2}}\left(v\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2\underset{xy\notin {G}_{1},x=y}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2}}{\sum }{d}_{{G}_{1},u=v}\left(x\right){d}_{{G}_{2}}\left(u\right)-2\underset{xy\notin {G}_{1},x=y}{\sum }\text{ }\text{ }\underset{uv\notin {G}_{2},u=v}{\sum }{d}_{{G}_{1}}\left(y\right){d}_{{G}_{2}}\left(v\right)\\ ={n}_{2}\left(\underset{xy\notin {G}_{1},x=y}{\sum }\left[{d}_{{G}_{1}}\left(x\right)+{d}_{{G}_{2}}\left(y\right)\right]\right)\left(\underset{uv\notin {G}_{2},u=v}{\sum }1\right)+{n}_{1}\left(\underset{xy\notin {G}_{1},x=y}{\sum }1\right)\left(\underset{uv\notin {G}_{2},u=v}{\sum }\left[{d}_{{G}_{2}}\left(u\right)+{d}_{{G}_{2}}\left(v\right)\right]\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2\left(\underset{xy\notin {G}_{1},x=y}{\sum }{d}_{{G}_{1}}\left(x\right)\right)\left(\underset{uv\notin {G}_{2},u=v}{\sum }{d}_{{G}_{2}}\left(u\right)\right)-2\left(\underset{xy\notin {G}_{1},x=y}{\sum }{d}_{{G}_{1}}\left(y\right)\right)\left(\underset{uv\notin {G}_{2},u=v}{\sum }{d}_{{G}_{2}}\left(v\right)\right)\\ =4{n}_{2}^{2}{m}_{1}+4{n}_{1}^{2}{m}_{2}-16{m}_{1}{m}_{2}\end{array}$

$\begin{array}{l}{\left[D{D}^{*}\left({G}_{1}\oplus {G}_{2}\right)\right]}^{2}\\ \le \left\{\frac{1}{{n}_{1}{n}_{2}\left({n}_{1}{n}_{2}-1\right)}\left[2{n}_{2}{m}_{2}\left(2{\stackrel{¯}{M}}_{1}\left({G}_{1}\right)+4{m}_{1}\right)+2{n}_{1}{M}_{1}\left({G}_{2}\right)\left(2{\stackrel{¯}{m}}_{1}+{n}_{1}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-4{M}_{1}\left({G}_{2}\right)\left(2{\stackrel{¯}{m}}_{1}\left({n}_{1}-1\right)+2{m}_{1}-{M}_{1}\left({\stackrel{¯}{G}}_{1}\right)\right)+2{n}_{2}{M}_{1}\left({G}_{1}\right)\left(2{\stackrel{¯}{m}}_{2}+{n}_{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+2{n}_{1}{m}_{1}\left(2{\stackrel{¯}{M}}_{1}\left({G}_{2}\right)+4{m}_{2}\right)-4{M}_{1}\left({G}_{1}\right)\left(2\left({n}_{2}-1\right){\stackrel{¯}{m}}_{2}+2{m}_{2}-{M}_{1}\left({\stackrel{¯}{G}}_{2}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+4{n}_{2}{M}_{1}\left({G}_{1}\right){m}_{2}+4{n}_{1}{m}_{1}{M}_{1}\left({G}_{2}\right)-4{M}_{1}\left({G}_{1}\right){M}_{1}\left({G}_{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\left[{n}_{2}\left(2{\stackrel{¯}{M}}_{1}\left({G}_{1}\right)+4{m}_{1}\right)\left(2{m}_{2}+{n}_{2}\right)+{n}_{1}\left(2{\stackrel{¯}{M}}_{1}\left({G}_{2}\right)+4{m}_{2}\right)\left(2{m}_{1}+{n}_{1}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-4\left(2{\stackrel{¯}{m}}_{1}\left({n}_{1}-1\right)+2{m}_{1}-{M}_{1}\left({\stackrel{¯}{G}}_{1}\right)\right)\left(2{\stackrel{¯}{m}}_{2}\left({n}_{2}-1\right)+2{m}_{2}-{M}_{1}\left({\stackrel{¯}{G}}_{2}\right)\right)\right]\\ \text{\hspace{0.17em}}{\begin{array}{c}\\ \end{array}-2\left(4{n}_{2}^{2}{m}_{1}+4{n}_{1}^{2}{m}_{2}-16{m}_{1}{m}_{2}\right)\right]\right\}}^{{n}_{1}{n}_{2}\left({n}_{1}{n}_{2}-1\right)}\end{array}$

Lemma 6.2.

$D{D}^{*}\left[{K}_{m}\oplus {K}_{1}\right]={\left(2m-2\right)}^{\frac{m\left(m-1\right)}{2}}$

Proof: Clearly the graph ${K}_{m}\oplus {K}_{1}$ is the complete graph ${K}_{m}$

$D{D}^{*}\left[{K}_{m}\oplus {K}_{1}\right]=D{D}^{*}{K}_{m}={\left(2m-2\right)}^{\frac{m\left(m-1\right)}{2}}$ (7)

Remark 6.3. Let ${G}_{1}={K}_{m}$ and ${G}_{2}={K}_{1}$ . We get

${n}_{1}=m,\text{ }\text{ }\text{ }\text{ }{n}_{2}=1,\text{ }\text{ }{m}_{1}=\frac{m\left(m-1\right)}{2},\text{ }\text{ }\text{ }\text{ }{m}_{2}=0,\text{ }\text{ }\text{ }\text{ }{\stackrel{¯}{m}}_{1}=0,\text{ }\text{ }\text{ }\text{ }{\stackrel{¯}{m}}_{2}=0$

${M}_{1}\left({G}_{1}\right)={M}_{1}\left({K}_{m}\right)=m{\left(m-1\right)}^{2},\text{ }\text{ }{M}_{1}\left({G}_{2}\right)={M}_{1}\left({K}_{1}\right)=0$

${M}_{1}\left({\stackrel{¯}{G}}_{1}\right)={M}_{1}\left({\stackrel{¯}{K}}_{m}\right)=0,\text{ }\text{ }{M}_{1}\left({\stackrel{¯}{G}}_{2}\right)={M}_{1}\left({\stackrel{¯}{K}}_{1}\right)=0$

${\stackrel{¯}{M}}_{1}\left({G}_{1}\right)={\stackrel{¯}{M}}_{1}\left({K}_{m}\right)=0,\text{ }\text{ }{\stackrel{¯}{M}}_{1}\left({G}_{2}\right)={\stackrel{¯}{M}}_{1}\left({K}_{1}\right)=0$

$\therefore$ In Theorem 6.1, put ${G}_{1}={K}_{m}$ and ${G}_{2}={K}_{1}$ , we get

$D{D}^{*}\left[{K}_{m}\oplus {K}_{1}\right]\le {\left(2m-2\right)}^{\frac{m\left(m-1\right)}{2}}$ (8)

From (7) and (8) our bound is tight.

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

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