# The multiplicative Zagreb indices of graph operations

- Kinkar C Das
^{1}, - Aysun Yurttas
^{2}, - Muge Togan
^{2}, - Ahmet Sinan Cevik
^{3}and - Ismail Naci Cangul
^{2}Email author

**2013**:90

https://doi.org/10.1186/1029-242X-2013-90

© Das et al.; licensee Springer 2013

**Received: **19 December 2012

**Accepted: **15 February 2013

**Published: **5 March 2013

## Abstract

Recently, Todeschini *et al.* (Novel Molecular Structure Descriptors - Theory and Applications I, pp. 73-100, 2010), Todeschini and Consonni (MATCH Commun. Math. Comput. Chem. 64:359-372, 2010) have proposed the multiplicative variants of ordinary Zagreb indices, which are defined as follows:

These two graph invariants are called *multiplicative Zagreb indices* by Gutman (Bull. Soc. Math. Banja Luka 18:17-23, 2011). In this paper the upper bounds on the multiplicative Zagreb indices of the join, Cartesian product, corona product, composition and disjunction of graphs are derived and the indices are evaluated for some well-known graphs.

**MSC:**05C05, 05C90, 05C07.

## Keywords

## 1 Introduction

*G*is a graph with a vertex set $V(G)$ and an edge set $E(G)$. For a graph

*G*, the degree of a vertex

*v*is the number of edges incident to

*v*and is denoted by ${d}_{G}(v)$. A topological index $Top(G)$ of a graph

*G*is a number with the property that for every graph

*H*isomorphic to

*G*, $Top(H)=Top(G)$. Recently, Todeschini

*et al.*[1, 2] have proposed the multiplicative variants of ordinary Zagreb indices, which are defined as follows:

Mathematical properties and applications of multiplicative Zagreb indices are reported in [1–6]. Mathematical properties and applications of multiplicative sum Zagreb indices are reported in [7]. For other undefined notations and terminology from graph theory, the readers are referred to [8].

In [9, 10], Khalifeh *et al.* computed some exact formulae for the hyper-Wiener index and Zagreb indices of the join, Cartesian product, composition, disjunction and symmetric difference of graphs. Some more properties and applications of graph products can be seen in the classical book [11].

In this paper, we give some upper bounds for the multiplicative Zagreb index of various graph operations such as join, corona product, Cartesian product, composition, disjunction, *etc.* Moreover, computations are done for some well-known graphs.

## 2 Multiplicative Zagreb index of graph operations

We begin this section with two standard inequalities as follows.

**Lemma 1** (AM-GM inequality)

*Let*${x}_{1},{x}_{2},\dots ,{x}_{n}$

*be nonnegative numbers*.

*Then*

*holds with equality if and only if all the* ${x}_{k}$*’s are equal*.

**Lemma 2** (Weighted AM-GM inequality)

*Let*${x}_{1},{x}_{2},\dots ,{x}_{n}$

*be nonnegative numbers and also let*${w}_{1},{w}_{2},\dots ,{w}_{n}$

*be nonnegative weights*.

*Set*$w={w}_{1}+{w}_{2}+\cdots +{w}_{n}$.

*If*$w>0$,

*then the inequality*

*holds with equality if and only if all the* ${x}_{k}$ *with* ${w}_{k}>0$ *are equal*.

Let ${G}_{1}$ and ${G}_{2}$ be two graphs with ${n}_{1}$ and ${n}_{2}$ vertices and ${m}_{1}$ and ${m}_{2}$ edges, respectively. The join ${G}_{1}\vee {G}_{2}$ of graphs ${G}_{1}$ and ${G}_{2}$ with disjoint vertex sets $V({G}_{1})$ and $V({G}_{2})$ and edge sets $E({G}_{1})$ and $E({G}_{2})$ is the graph union ${G}_{1}\cup {G}_{2}$ together with all the edges joining $V({G}_{1})$ and $V({G}_{2})$. Thus, for example, ${\overline{K}}_{p}\vee {\overline{K}}_{q}={K}_{p,q}$, the complete bipartite graph. We have $|V({G}_{1}\vee {G}_{2})|={n}_{1}+{n}_{2}$ and $|E({G}_{1}\vee {G}_{2})|={m}_{1}+{m}_{2}+{n}_{1}{n}_{2}$.

**Theorem 1**

*Let*${G}_{1}$

*and*${G}_{2}$

*be two graphs*.

*Then*

*and*

*where* ${n}_{1}$ *and* ${n}_{2}$ *are the numbers of vertices of* ${G}_{1}$ *and* ${G}_{2}$, *and* ${m}_{1}$, ${m}_{2}$ *are the numbers of edges of* ${G}_{1}$ *and* ${G}_{2}$, *respectively*. *Moreover*, *the equality holds in* (3) *if and only if both* ${G}_{1}$ *and* ${G}_{2}$ *are regular graphs*, *that is*, ${G}_{1}\vee {G}_{2}$ *is a regular graph and the equality holds in* (4) *if and only if both* ${G}_{1}$ *and* ${G}_{2}$ *are regular graphs*, *that is*, ${G}_{1}\vee {G}_{2}$ *is a regular graph*.

*Proof*

That is, for ${u}_{i},{u}_{k}\in V({G}_{1})$ and ${v}_{j},{v}_{\ell}\in V({G}_{2})$, we get ${d}_{{G}_{1}}({u}_{i})={d}_{{G}_{1}}({u}_{k})$ and ${d}_{{G}_{2}}({v}_{j})={d}_{{G}_{2}}({v}_{\ell})$. Hence the equality holds in (3) if and only if both ${G}_{1}$ and ${G}_{2}$ are regular graphs, that is, ${G}_{1}\vee {G}_{2}$ is a regular graph.

Hence the equality holds in (4) if and only if both ${G}_{1}$ and ${G}_{2}$ are regular graphs, that is, ${G}_{1}\vee {G}_{2}$ is a regular graph. □

**Example 1**Consider two cycle graphs ${C}_{p}$ and ${C}_{q}$. We thus have

*Cartesian product*${G}_{1}\u22a0{G}_{2}$ of graphs ${G}_{1}$ and ${G}_{2}$ has the vertex set $V({G}_{1}\times {G}_{2})=V({G}_{1})\times V({G}_{2})$ and $({u}_{i},{v}_{j})({u}_{k},{v}_{\ell})$ is an edge of ${G}_{1}\u22a0{G}_{2}$ if

**Theorem 2**

*Let*${G}_{1}$

*and*${G}_{2}$

*be two connected graphs*.

*Then*

- (i)$\prod _{1}({G}_{1}\u22a0{G}_{2})\le {\left[\frac{{n}_{2}{M}_{1}({G}_{1})+{n}_{1}{M}_{1}({G}_{2})+8{m}_{1}{m}_{2}}{{n}_{1}{n}_{2}}\right]}^{{n}_{1}{n}_{2}}.$(6)

*The equality holds in*(6)

*if and only if*${G}_{1}\u22a0{G}_{2}$

*is a regular graph*.

- (ii)$\begin{array}{rcl}\prod _{2}({G}_{1}\u22a0{G}_{2})& \le & \frac{1}{{(2{n}_{1}{m}_{2})}^{2{n}_{1}{m}_{2}}}{({n}_{1}{M}_{1}({G}_{2})+4{m}_{1}{m}_{2})}^{2{n}_{1}{m}_{2}}\\ \times \frac{1}{{(2{n}_{2}{m}_{1})}^{2{n}_{2}{m}_{1}}}{({n}_{2}{M}_{1}({G}_{1})+4{m}_{1}{m}_{2})}^{2{n}_{2}{m}_{1}}.\end{array}$(7)

*Moreover*, *the equality holds in* (7) *if and only if* ${G}_{1}\u22a0{G}_{2}$ *is a regular graph*.

*Proof*

Moreover, the equality holds in (8) if and only if ${d}_{{G}_{1}}({u}_{i})+{d}_{{G}_{2}}({v}_{j})={d}_{{G}_{1}}({u}_{k})+{d}_{{G}_{2}}({v}_{\ell})$ for any $({u}_{i},{v}_{j}),({u}_{k},{v}_{\ell})\in V({G}_{1}\u22a0{G}_{2})$ by Lemma 1. Since both ${G}_{1}$ and ${G}_{2}$ are connected graphs, one can easily see that the equality holds in (8) if and only if ${d}_{{G}_{1}}({u}_{i})={d}_{{G}_{1}}({u}_{k})$, ${u}_{i},{u}_{k}\in V({G}_{1})$ and ${d}_{{G}_{2}}({v}_{j})={d}_{{G}_{2}}({v}_{\ell})$, ${v}_{j},{v}_{\ell}\in V({G}_{2})$. Hence the equality holds in (6) if and only if both ${G}_{1}$ and ${G}_{2}$ are regular graphs, that is, ${G}_{1}\u22a0{G}_{2}$ is a regular graph. This completes the first part of the proof.

Hence the second part of the proof is over.

The equality holds in (9) and (10) if and only if ${d}_{{G}_{2}}({v}_{j})={d}_{{G}_{2}}({v}_{\ell})$ for any ${v}_{j},{v}_{\ell}\in V({G}_{2})$ and ${d}_{{G}_{1}}({u}_{i})={d}_{{G}_{1}}({u}_{k})$ for any ${u}_{i},{u}_{k}\in V({G}_{1})$ by Lemmas 1 and 2. Hence the equality holds in (7) if and only if both ${G}_{1}$ and ${G}_{2}$ are regular graphs, that is, ${G}_{1}\u22a0{G}_{2}$ is a regular graph. This completes the proof. □

**Example 2**Consider a cycle graph ${C}_{p}$ and a complete graph ${K}_{q}$. We thus have

The *corona product* ${G}_{1}\circ {G}_{2}$ of two graphs ${G}_{1}$ and ${G}_{2}$ is defined to be the graph Γ obtained by taking one copy of ${G}_{1}$ (which has ${n}_{1}$ vertices) and ${n}_{1}$ copies of ${G}_{2}$, and then joining the *i* th vertex of ${G}_{1}$ to every vertex in the *i* th copy of ${G}_{2}$, $i=1,2,\dots ,{n}_{1}$.

Let ${G}_{1}=(V,E)$ and ${G}_{2}=(V,E)$ be two graphs such that $V({G}_{1})=\{{u}_{1},{u}_{2},\dots ,{u}_{{n}_{1}}\}$, $|E({G}_{1})|={m}_{1}$ and $V({G}_{2})=\{{v}_{1},{v}_{2},\dots ,{v}_{{n}_{2}}\}$, $|E({G}_{2})|={m}_{2}$. Then it follows from the definition of the corona product that ${G}_{1}\circ {G}_{2}$ has ${n}_{1}(1+{n}_{2})$ vertices and ${m}_{1}+{n}_{1}{m}_{2}+{n}_{1}{n}_{2}$ edges, where $V({G}_{1}\circ {G}_{2})=\{({u}_{i},{v}_{j}),i=1,2,\dots ,{n}_{1};j=0,1,2,\dots ,{n}_{2}\}$ and $E({G}_{1}\circ {G}_{2})=\{(({u}_{i},{v}_{0}),({u}_{k},{v}_{0})),({u}_{i},{u}_{k})\in E({G}_{1})\}\cup $ $\{(({u}_{i},{v}_{j}),({u}_{i},{v}_{\ell})),({v}_{j},{v}_{\ell})\in E({G}_{2}),i=1,2,\dots ,{n}_{1}\}\cup $ $\{(({u}_{i},{v}_{0}),({u}_{i},{v}_{\ell})),\ell =1,2,\dots ,{n}_{2},i=1,2,\dots ,{n}_{1}\}$. It is clear that if ${G}_{1}$ is connected, then ${G}_{1}\circ {G}_{2}$ is connected, and in general ${G}_{1}\circ {G}_{2}$ is not isomorphic to ${G}_{2}\circ {G}_{1}$.

**Theorem 3**

*The first and second multiplicative Zagreb indices of the corona product are computed as follows*:

- (i)$\prod _{1}({G}_{1}\circ {G}_{2})\le \frac{1}{{n}_{1}^{{n}_{1}}{n}_{2}^{{n}_{1}{n}_{2}}}{M}_{1}{({G}_{1})}^{{n}_{1}}{({M}_{1}({G}_{2})+4{m}_{2}+{n}_{2})}^{{n}_{1}{n}_{2}},$(11)
- (ii)$\begin{array}{rcl}\prod _{2}({G}_{1}\circ {G}_{2})& \le & {\left[\frac{{M}_{2}({G}_{1})+{n}_{2}{M}_{1}({G}_{1})+{n}_{2}^{2}}{{m}_{1}}\right]}^{{m}_{1}}{\left[\frac{{M}_{2}({G}_{2})+{M}_{1}({G}_{2})+1}{{m}_{2}}\right]}^{{n}_{1}{m}_{2}}\\ \times {\left[\frac{4{m}_{1}{m}_{2}+{n}_{1}{n}_{2}^{2}+2{m}_{1}{n}_{2}+2{m}_{2}{n}_{1}{n}_{2}}{{n}_{1}{n}_{2}}\right]}^{{n}_{1}{n}_{2}},\end{array}$(12)

*where* ${M}_{1}({G}_{i})$ *and* ${M}_{2}({G}_{i})$ *are the first and second Zagreb indices of* ${G}_{i}$, *where* $i=1,2$, *respectively*. *Moreover*, *both equalities in* (11) *and* (12) *hold if and only if* ${G}_{1}\circ {G}_{2}$ *is a regular graph*.

*Proof*

The equality holds in (13) if and only if ${d}_{{G}_{1}}({u}_{i})={d}_{{G}_{1}}({u}_{k})$, ${u}_{i},{u}_{k}\in V({G}_{1})$ and ${d}_{{G}_{2}}({v}_{j})={d}_{{G}_{2}}({v}_{\ell})$, ${v}_{j},{v}_{\ell}\in V({G}_{2})$, that is, both ${G}_{1}$ and ${G}_{2}$ are regular graphs, that is, ${G}_{1}\circ {G}_{2}$ is a regular graph.

The above equality holds if and only if ${d}_{{G}_{1}}({u}_{i})={d}_{{G}_{1}}({u}_{k})$ for any ${u}_{i},{u}_{k}\in V({G}_{1})$ and ${d}_{{G}_{2}}({v}_{j})={d}_{{G}_{2}}({v}_{\ell})$ for any ${v}_{j},{v}_{\ell}\in V({G}_{2})$, that is, both ${G}_{1}$ and ${G}_{2}$ are regular graphs, which implies that ${G}_{1}\circ {G}_{2}$ is a regular graph. This completes the proof. □

**Example 3** ${\prod}_{1}({C}_{p}\circ {K}_{q})={q}^{2pq}{(q+2)}^{2p}$ and ${\prod}_{2}({C}_{p}\circ {K}_{q})={q}^{p{q}^{2}}{(q+2)}^{p(q+2)}$.

*composition*(also called

*lexicographic product*[12]) $G={G}_{1}[{G}_{2}]$ of graphs ${G}_{1}$ and ${G}_{2}$ with disjoint vertex sets $V({G}_{1})$ and $V({G}_{2})$ and edge sets $E({G}_{1})$ and $E({G}_{2})$ is the graph with a vertex set $V({G}_{1})\times V({G}_{2})$ and $({u}_{i},{v}_{j})$ is adjacent to $({u}_{k},{v}_{\ell})$ whenever

**Theorem 4**

*The first and second multiplicative Zagreb indices of the composition*${G}_{1}[{G}_{2}]$

*of graphs*${G}_{1}$

*and*${G}_{2}$

*are bounded above as follows*:

- (i)$\prod _{1}({G}_{1}[{G}_{2}])\le \frac{1}{{({n}_{1}{n}_{2})}^{{n}_{1}{n}_{2}}}{[{n}_{2}^{3}{M}_{1}({G}_{1})+8{n}_{2}{m}_{1}{m}_{2}+{n}_{1}{M}_{1}({G}_{2})]}^{{n}_{1}{n}_{2}},$(14)
- (ii)$\begin{array}{rcl}\prod _{2}({G}_{1}[{G}_{2}])& \le & \frac{1}{{({n}_{1}{m}_{2})}^{{n}_{1}{m}_{2}}}{[{m}_{2}{n}_{2}^{2}{M}_{1}({G}_{1})+2{n}_{2}{m}_{1}{M}_{1}({G}_{2})+{n}_{1}{M}_{2}({G}_{2})]}^{{n}_{1}{m}_{2}}\\ \times \frac{1}{{({n}_{2}{m}_{1})}^{{m}_{1}{n}_{2}^{2}}}{[{n}_{2}^{3}{M}_{2}({G}_{1})+{m}_{1}{M}_{1}({G}_{2})+2{m}_{2}{n}_{2}{M}_{1}({G}_{1})]}^{{n}_{2}^{2}{m}_{1}},\end{array}$(15)

*where* ${M}_{1}({G}_{i})$ *and* ${M}_{2}({G}_{i})$ *are the first and second Zagreb indices of* ${G}_{i}$, *where* $i=1,2$. *Moreover*, *the equalities in* (14) *and* (15) *hold if and only if* ${G}_{1}\circ {G}_{2}$ *is a regular graph*.

*Proof*

The equality holds in (16) if and only if ${d}_{{G}_{1}}({u}_{i})={d}_{{G}_{1}}({u}_{k})$, ${u}_{i},{u}_{k}\in V({G}_{1})$ and ${d}_{{G}_{2}}({v}_{j})={d}_{{G}_{2}}({v}_{\ell})$, ${v}_{j},{v}_{\ell}\in V({G}_{2})$ (by Lemma 1), that is, both ${G}_{1}$ and ${G}_{2}$ are regular graphs, that is, ${G}_{1}\circ {G}_{2}$ is a regular graph.

which gives the required result in (15).

The equality holds in (17) and (18) if and only if ${d}_{{G}_{1}}({u}_{i})={d}_{{G}_{1}}({u}_{k})$, ${u}_{i},{u}_{k}\in V({G}_{1})$ and ${d}_{{G}_{2}}({v}_{j})={d}_{{G}_{2}}({v}_{\ell})$, ${v}_{j},{v}_{\ell}\in V({G}_{2})$ (by Lemma 1), that is, both ${G}_{1}$ and ${G}_{2}$ are regular graphs, that is, ${G}_{1}\circ {G}_{2}$ is a regular graph. □

**Example 4** ${\prod}_{1}({C}_{p}[{C}_{q}])={2}^{2pq}{(q+1)}^{2pq}$ and ${\prod}_{2}({C}_{p}[{C}_{q}])={2}^{2pq(q+1)}{(q+1)}^{2pq(q+1)}$.

The *disjunction* ${G}_{1}\otimes {G}_{2}$ of graphs ${G}_{1}$ and ${G}_{2}$ is the graph with a vertex set $V({G}_{1})\times V({G}_{2})$ and $({u}_{i},{v}_{j})$ is adjacent to $({u}_{k},{v}_{\ell})$ whenever ${u}_{i}{u}_{k}\in E({G}_{1})$ or ${v}_{j}{v}_{\ell}\in E({G}_{2})$.

**Theorem 5**

*The first and second multiplicative Zagreb indices of the disjunction are computed as follows*:

- (i)$\begin{array}{rcl}\prod _{1}({G}_{1}\otimes {G}_{2})& \le & \frac{1}{{({n}_{1}{n}_{2})}^{{n}_{1}{n}_{2}}}[{n}_{2}^{3}{M}_{1}({G}_{1})+{n}_{1}^{3}{M}_{1}({G}_{2})+{M}_{1}({G}_{1}){M}_{1}({G}_{2})\\ {+8{n}_{1}{n}_{2}{m}_{1}{m}_{2}-4{n}_{1}{m}_{1}{M}_{1}({G}_{2})-4{n}_{2}{m}_{2}{M}_{1}({G}_{1})]}^{{n}_{1}{n}_{2}},\end{array}$(19)
- (ii)

*where* $Q={\sum}_{{u}_{i}\in V({G}_{1})}{\sum}_{{v}_{j}\in V({G}_{2})}P=2({n}_{2}^{2}{m}_{1}+{n}_{1}^{2}{m}_{2}-2{m}_{1}{m}_{2})$ *and* ${M}_{1}({G}_{i})$ *is the first Zagreb index of* ${G}_{i}$, $i=1,2$. *Moreover*, *the equalities in* (19) *and* (20) *hold if and only if* ${G}_{1}\circ {G}_{2}$ *is a regular graph*.

*Proof*We have ${d}_{{G}_{1}\otimes {G}_{2}}({u}_{i},{v}_{j})={n}_{2}{d}_{{G}_{1}}({u}_{i})+{n}_{1}{d}_{{G}_{2}}({v}_{j})-{d}_{{G}_{1}}({u}_{i}){d}_{{G}_{2}}({v}_{j})$. By the definition of the first multiplicative Zagreb index, we have

The equality holds in (21) if and only if ${d}_{{G}_{1}}({u}_{i})={d}_{{G}_{1}}({u}_{k})$, ${u}_{i},{u}_{k}\in V({G}_{1})$ and ${d}_{{G}_{2}}({v}_{j})={d}_{{G}_{2}}({v}_{\ell})$, ${v}_{j},{v}_{\ell}\in V({G}_{2})$ (by Lemma 1), that is, both ${G}_{1}$ and ${G}_{2}$ are regular graphs, that is, ${G}_{1}\circ {G}_{2}$ is a regular graph.

Hence the first part of the proof is over.

The equality holds in (22) if and only if ${d}_{{G}_{1}}({u}_{i})={d}_{{G}_{1}}({u}_{k})$, where ${u}_{i},{u}_{k}\in V({G}_{1})$ and ${d}_{{G}_{2}}({v}_{j})={d}_{{G}_{2}}({v}_{\ell})$, where ${v}_{j},{v}_{\ell}\in V({G}_{2})$ (by Lemma 1), that is, both ${G}_{1}$ and ${G}_{2}$ are regular graphs, and so the graph ${G}_{1}\circ {G}_{2}$ is regular. □

**Example 5** ${\prod}_{1}({K}_{p}\otimes {C}_{q})={(pq-q+2)}^{2pq}$ and ${\prod}_{2}({K}_{p}\otimes {C}_{q})={(pq-q+2)}^{pq(pq-q+2)}$.

*symmetric difference*${G}_{1}\oplus {G}_{2}$ of two graphs ${G}_{1}$ and ${G}_{2}$ is the graph with a vertex set $V({G}_{1})\times V({G}_{2})$ in which $({u}_{i},{v}_{j})$ is adjacent to $({u}_{k},{v}_{\ell})$ whenever ${u}_{i}$ is adjacent to ${u}_{k}$ in ${G}_{1}$ or ${v}_{i}$ is adjacent to ${v}_{\ell}$ in ${G}_{2}$, but not both. The degree of a vertex $({u}_{i},{v}_{j})$ of ${G}_{1}\oplus {G}_{2}$ is given by

while the number of edges in ${G}_{1}\oplus {G}_{2}$ is ${n}_{1}^{2}{m}_{2}+{n}_{2}^{2}{m}_{1}-4{m}_{1}{m}_{2}$.

**Theorem 6**

*The first and second multiplicative Zagreb indices of the symmetric difference*${G}_{1}\oplus {G}_{2}$

*of two graphs*${G}_{1}$

*and*${G}_{2}$

*are bounded above as follows*:

- (i)$\begin{array}{rcl}\prod _{1}({G}_{1}\oplus {G}_{2})& \le & \frac{1}{{({n}_{1}{n}_{2})}^{{n}_{1}{n}_{2}}}[{n}_{2}^{3}{M}_{1}({G}_{1})+{n}_{1}^{3}{M}_{1}({G}_{2})+4{M}_{1}({G}_{1}){M}_{1}({G}_{2})\\ {+8{n}_{1}{n}_{2}{m}_{1}{m}_{2}-8{n}_{1}{m}_{1}{M}_{1}({G}_{2})-8{n}_{2}{m}_{2}{M}_{1}({G}_{1})]}^{{n}_{1}{n}_{2}},\end{array}$(23)
- (ii)

*where* $Q={\sum}_{{u}_{i}\in V({G}_{1})}{\sum}_{{v}_{j}\in V({G}_{2})}P=2({n}_{2}^{2}{m}_{1}+{n}_{1}^{2}{m}_{2}-4{m}_{1}{m}_{2})$ *and* ${M}_{1}({G}_{i})$ *is the first Zagreb index of* ${G}_{i}$, *for* $i=1,2$. *Moreover*, *the equalities in* (23) *and* (24) *hold if and only if* ${G}_{1}\circ {G}_{2}$ *is a regular graph*.

*Proof*

The equality holds in (25) if and only if ${d}_{{G}_{1}}({u}_{i})={d}_{{G}_{1}}({u}_{k})$, ${u}_{i},{u}_{k}\in V({G}_{1})$ and ${d}_{{G}_{2}}({v}_{j})={d}_{{G}_{2}}({v}_{\ell})$, ${v}_{j},{v}_{\ell}\in V({G}_{2})$ (by Lemma 1), that is, both ${G}_{1}$ and ${G}_{2}$ are regular graphs, which implies that ${G}_{1}\circ {G}_{2}$ is a regular graph.

where $P={n}_{2}{d}_{{G}_{1}}({u}_{i})+{n}_{1}{d}_{{G}_{2}}({v}_{j})-2{d}_{{G}_{1}}({u}_{i}){d}_{{G}_{2}}({v}_{j})$.

where $Q={\sum}_{{u}_{i}\in V({G}_{1})}{\sum}_{{v}_{j}\in V({G}_{2})}P=2({n}_{2}^{2}{m}_{1}+{n}_{1}^{2}{m}_{2}-4{m}_{1}{m}_{2})$. First part of the proof is over.

The equality holds in (26) if and only if ${d}_{{G}_{1}}({u}_{i})={d}_{{G}_{1}}({u}_{k})$, ${u}_{i},{u}_{k}\in V({G}_{1})$ and ${d}_{{G}_{2}}({v}_{j})={d}_{{G}_{2}}({v}_{\ell})$, ${v}_{j},{v}_{\ell}\in V({G}_{2})$ (by Lemma 1), that is, both ${G}_{1}$ and ${G}_{2}$ are regular graphs, which implies that ${G}_{1}\circ {G}_{2}$ is a regular graph. □

**Example 6** ${\prod}_{1}({G}_{1}\oplus {G}_{2})={(p+q-2)}^{2pq}$ and ${\prod}_{2}({G}_{1}\oplus {G}_{2})={(p+q-2)}^{pq(p+q-2)}$.

## Declarations

### Acknowledgements

Dedicated to Professor Hari M Srivastava.

All authors except the first one are partially supported by Research Project Offices of Uludağ (2012-15 and 2012-19) and Selçuk Universities. K.C. Das thanks for support the Sungkyunkwan University BK21 Project, BK21 Math Modeling HRD Div. Sungkyunkwan University, Suwon, Republic of Korea.

## Authors’ Affiliations

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