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The multiplicative Zagreb indices of graph operations
Journal of Inequalities and Applications volume 2013, Article number: 90 (2013)
Abstract
Recently, Todeschini et al. (Novel Molecular Structure Descriptors  Theory and Applications I, pp. 73100, 2010), Todeschini and Consonni (MATCH Commun. Math. Comput. Chem. 64:359372, 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 18:1723, 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 wellknown graphs.
MSC:05C05, 05C90, 05C07.
1 Introduction
Throughout this paper, we consider simple graphs which are finite, indirected graphs without loops and multiple edges. Suppose 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 hyperWiener 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 wellknown graphs.
2 Multiplicative Zagreb index of graph operations
We begin this section with two standard inequalities as follows.
Lemma 1 (AMGM 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 AMGM 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
Now,
and by (1) this above equality is actually less than or equal to
Moreover, the above equality holds if and only if
and
(by Lemma 1), that is, for {u}_{i},{u}_{k}\in V({G}_{1}) and {v}_{j},{v}_{\ell}\in V({G}_{2}),
and
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.
Now, since
we then obtain
and by (1)
However, from the last inequality, we get
Furthermore, for both connected graphs {G}_{1} and {G}_{2}, the equality holds in (5) iff
for any {u}_{i}{u}_{r},{u}_{i}{u}_{k}\in E({G}_{1}); and
for any {v}_{j}{v}_{r},{v}_{j}{v}_{\ell}\in E({G}_{2}) as well as
for any {u}_{i}\in V({G}_{1}), {v}_{j},{v}_{\ell}\in V({G}_{2}); and
for any {v}_{j}\in V({G}_{2}), {u}_{i},{u}_{k}\in V({G}_{1}) by Lemma 1. Thus one can easily see that the equality holds in (5) if and only if for {u}_{i},{u}_{k}\in V({G}_{1}) and {v}_{j},{v}_{\ell}\in V({G}_{2}),
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
The 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
By the definition of the first multiplicative Zagreb index, we have
On the other hand, by (1)
But as {\sum}_{{u}_{i}\in V({G}_{1})}{d}_{{G}_{1}}{({u}_{i})}^{2}={M}_{1}({G}_{1}) and {\sum}_{{v}_{j}\in V({G}_{2})}{d}_{{G}_{2}}{({v}_{j})}^{2}={M}_{1}({G}_{2}), the last statement in (8) is less than or equal to
which equals to
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.
By the definition of the second multiplicative Zagreb index, we have
This actually can be written as
or, equivalently,
After that, by (2) we get
Moreover, since
By (1) the final statement in (9) becomes
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
By the definition of the first multiplicative Zagreb index, we have
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.
By the definition of the second multiplicative Zagreb index, we have
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)}.
The 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
By the definition of the first multiplicative Zagreb index, we have
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.
By the definition of the second multiplicative Zagreb index, we have
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)
(20)
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.
By the definition of the second multiplicative Zagreb index, we have
where
Using the weighted arithmeticgeometric mean inequality in (2), {\prod}_{2}({G}_{1}\otimes {G}_{2}) is less than or equal to
where
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})={(pqq+2)}^{2pq} and {\prod}_{2}({K}_{p}\otimes {C}_{q})={(pqq+2)}^{pq(pqq+2)}.
The 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)
(24)
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
We have
By the definition of the first multiplicative Zagreb index, we have
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.
By the definition of the second multiplicative Zagreb index, we have
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}).
Using the weighted arithmeticgeometric mean inequality in (2), we get
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+q2)}^{2pq} and {\prod}_{2}({G}_{1}\oplus {G}_{2})={(p+q2)}^{pq(p+q2)}.
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Acknowledgements
Dedicated to Professor Hari M Srivastava.
All authors except the first one are partially supported by Research Project Offices of Uludağ (201215 and 201219) 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.
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Das, K.C., Yurttas, A., Togan, M. et al. The multiplicative Zagreb indices of graph operations. J Inequal Appl 2013, 90 (2013). https://doi.org/10.1186/1029242X201390
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DOI: https://doi.org/10.1186/1029242X201390
Keywords
 graph
 multiplicative Zagreb index
 graph operations