The multiplicative Zagreb indices of graph operations

Das, Kinkar C; Yurttas, Aysun; Togan, Muge; Cevik, Ahmet Sinan; Cangul, Ismail Naci

doi:10.1186/1029-242X-2013-90

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Published: 05 March 2013

The multiplicative Zagreb indices of graph operations

Kinkar C Das¹,
Aysun Yurttas²,
Muge Togan²,
Ahmet Sinan Cevik³ &
…
Ismail Naci Cangul²

Journal of Inequalities and Applications volume 2013, Article number: 90 (2013) Cite this article

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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:

\prod_{1} = \prod_{1} (G) = \prod_{v \in V (G)} d_{G} {(v)}^{2}, \prod_{2} = \prod_{2} (G) = \prod_{u v \in E (G)} d_{G} (u) d_{G} (v) .

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.

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:

\prod_{1} = \prod_{1} (G) = \prod_{v \in V (G)} d_{G} {(v)}^{2}, \prod_{2} = \prod_{2} (G) = \prod_{u v \in E (G)} d_{G} (u) d_{G} (v) .

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

\frac{x_{1} + x_{2} + \dots + x_{n}}{n} \geq \sqrt[n]{x_{1} x_{2} \dots x_{n}}

(1)

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 $w = w_{1} + w_{2} + \dots + w_{n}$ . If $w > 0$ , then the inequality

\frac{w_{1} x_{1} + w_{2} x_{2} + \dots + w_{n} x_{n}}{w} \geq \sqrt[w]{x_{1}^{w_{1}} x_{2}^{w_{2}} \dots x_{n}^{w_{n}}}

(2)

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 $G_{1} \lor G_{2}$ 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, ${\bar{K}}_{p} \lor {\bar{K}}_{q} = K_{p, q}$ , the complete bipartite graph. We have $| V (G_{1} \lor G_{2}) | = n_{1} + n_{2}$ and $| E (G_{1} \lor G_{2}) | = m_{1} + m_{2} + n_{1} n_{2}$ .

Theorem 1 Let $G_{1}$ and $G_{2}$ be two graphs. Then

\prod_{1} (G_{1} \lor G_{2}) \leq {[\frac{M_{1} (G_{1}) + 4 m_{1} n_{2} + n_{1} n_{2}^{2}}{n_{1}}]}^{n_{1}} \times {[\frac{M_{1} (G_{2}) + 4 n_{1} m_{2} + n_{2} n_{1}^{2}}{n_{2}}]}^{n_{2}}

(3)

and

\begin{array}{rcl} \prod_{2} (G_{1} \lor G_{2}) & \leq & {[\frac{M_{2} (G_{1}) + n_{2} M_{1} (G_{1}) + m_{1} n_{2}^{2}}{m_{1}}]}^{m_{1}} \times {[\frac{M_{2} (G_{2}) + n_{1} M_{1} (G_{2}) + m_{2} n_{1}^{2}}{m_{2}}]}^{m_{2}} \\ \times {[\frac{4 m_{1} m_{2} + 2 n_{1} n_{2} (m_{1} + m_{2}) + {(n_{1} n_{2})}^{2}}{n_{1} n_{2}}]}^{n_{1} n_{2}}, \end{array}

(4)

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, $G_{1} \lor G_{2}$ 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, $G_{1} \lor G_{2}$ is a regular graph.

Proof

Now,

\begin{array}{rcl} \prod_{1} (G_{1} \lor G_{2}) & = & \prod_{(u_{i}, v_{j}) \in V (G_{1} \lor G_{2})} d_{G_{1} \lor G_{2}} {(u_{i}, v_{j})}^{2} \\ = & \prod_{u_{i} \in V (G_{1})} {(d_{G_{1}} (u_{i}) + n_{2})}^{2} \prod_{v_{j} \in V (G_{2})} {(d_{G_{2}} (v_{j}) + n_{1})}^{2} \\ = & \prod_{u_{i} \in V (G_{1})} (d_{G_{1}} {(u_{i})}^{2} + 2 n_{2} d_{G_{1}} (u_{i}) + n_{2}^{2}) \prod_{v_{j} \in V (G_{2})} (d_{G_{2}} {(v_{j})}^{2} + 2 n_{1} d_{G_{2}} (v_{j}) + n_{1}^{2}) \end{array}

and by (1) this above equality is actually less than or equal to

Moreover, the above equality holds if and only if

d_{G_{1}} {(u_{i})}^{2} + 2 n_{2} d_{G_{1}} (u_{i}) + n_{2}^{2} = d_{G_{1}} {(u_{k})}^{2} + 2 n_{2} d_{G_{1}} (u_{k}) + n_{2}^{2} (u_{i}, u_{k} \in V (G_{1}))

and

d_{G_{2}} {(v_{j})}^{2} + 2 n_{1} d_{G_{2}} (v_{j}) + n_{1}^{2} = d_{G_{2}} {(v_{ℓ})}^{2} + 2 n_{1} d_{G_{2}} (v_{ℓ}) + n_{1}^{2} (v_{j}, v_{ℓ} \in V (G_{2}))

(by Lemma 1), that is, for $u_{i}, u_{k} \in V (G_{1})$ and $v_{j}, v_{ℓ} \in V (G_{2})$ ,

(d_{G_{1}} (u_{i}) - d_{G_{1}} (u_{k})) (d_{G_{1}} (u_{i}) + d_{G_{1}} (u_{k}) + 2 n_{2})

and

(d_{G_{2}} (v_{j}) - d_{G_{2}} (v_{ℓ})) (d_{G_{2}} (v_{j}) + d_{G_{2}} (v_{ℓ}) + 2 n_{1}) .

That is, for $u_{i}, u_{k} \in V (G_{1})$ and $v_{j}, v_{ℓ} \in V (G_{2})$ , we get $d_{G_{1}} (u_{i}) = d_{G_{1}} (u_{k})$ and $d_{G_{2}} (v_{j}) = d_{G_{2}} (v_{ℓ})$ . Hence the equality holds in (3) if and only if both $G_{1}$ and $G_{2}$ are regular graphs, that is, $G_{1} \lor G_{2}$ is a regular graph.

Now, since

\prod_{2} (G_{1} \lor G_{2}) = \prod_{(u_{i}, v_{j}) (u_{k}, v_{ℓ}) \in E (G_{1} \lor G_{2})} d_{G_{1} \lor G_{2}} (u_{i}, v_{j}) d_{G_{1} \lor G_{2}} (u_{k}, v_{ℓ}),

we then obtain

and by (1)

(5)

However, from the last inequality, we get

Furthermore, for both connected graphs $G_{1}$ and $G_{2}$ , the equality holds in (5) iff

d_{G_{1}} (u_{i}) d_{G_{1}} (u_{r}) + n_{2} (d_{G_{1}} (u_{i}) + d_{G_{1}} (u_{r})) + n_{2}^{2} = d_{G_{1}} (u_{i}) d_{G_{1}} (u_{k}) + n_{2} (d_{G_{1}} (u_{i}) + d_{G_{1}} (u_{k})) + n_{2}^{2}

for any $u_{i} u_{r}, u_{i} u_{k} \in E (G_{1})$ ; and

d_{G_{2}} (v_{j}) d_{G_{2}} (v_{r}) + n_{1} (d_{G_{2}} (v_{j}) + d_{G_{2}} (v_{r})) + n_{1}^{2} = d_{G_{2}} (v_{j}) d_{G_{2}} (v_{ℓ}) + n_{1} (d_{G_{2}} (v_{j}) + d_{G_{2}} (v_{ℓ})) + n_{1}^{2}

for any $v_{j} v_{r}, v_{j} v_{ℓ} \in E (G_{2})$ as well as

for any $u_{i} \in V (G_{1})$ , $v_{j}, v_{ℓ} \in V (G_{2})$ ; 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 $v_{j}, v_{ℓ} \in V (G_{2})$ ,

d_{G_{1}} (u_{i}) = d_{G_{1}} (u_{k}) and d_{G_{2}} (v_{j}) = d_{G_{2}} (v_{ℓ}) .

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

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

\prod_{1} (C_{p} \lor C_{q}) = {(p + 2)}^{2 q} {(q + 2)}^{2 p} and \prod_{2} (C_{p} \lor C_{q}) = {(p + 2)}^{(p + 2) q} {(q + 2)}^{(q + 2) p} .

The Cartesian product $G_{1} ⊠ G_{2}$ 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 $(u_{i}, v_{j}) (u_{k}, v_{ℓ})$ is an edge of $G_{1} ⊠ G_{2}$ if

Theorem 2 Let $G_{1}$ and $G_{2}$ be two connected graphs. Then

(i)
$\prod_{1} (G_{1} ⊠ G_{2}) \leq {[\frac{n_{2} M_{1} (G_{1}) + n_{1} M_{1} (G_{2}) + 8 m_{1} m_{2}}{n_{1} n_{2}}]}^{n_{1} n_{2}} .$
(6)

The equality holds in (6) if and only if $G_{1} ⊠ G_{2}$ is a regular graph.

(ii)
$\begin{array}{rcl} \prod_{2} (G_{1} ⊠ G_{2}) & \leq & \frac{1}{{(2 n_{1} m_{2})}^{2 n_{1} m_{2}}} {(n_{1} M_{1} (G_{2}) + 4 m_{1} m_{2})}^{2 n_{1} m_{2}} \\ \times \frac{1}{{(2 n_{2} m_{1})}^{2 n_{2} m_{1}}} {(n_{2} M_{1} (G_{1}) + 4 m_{1} m_{2})}^{2 n_{2} m_{1}} . \end{array}$
(7)

Moreover, the equality holds in (7) if and only if $G_{1} ⊠ G_{2}$ is a regular graph.

Proof

By the definition of the first multiplicative Zagreb index, we have

\begin{array}{rcl} \prod_{1} (G_{1} ⊠ G_{2}) & = & \prod_{(u_{i}, v_{j}) \in V (G_{1} ⊠ G_{2})} {(d_{G_{1}} (u_{i}) + d_{G_{2}} (v_{j}))}^{2} \\ = & \prod_{u_{i} \in V (G_{1})} \prod_{v_{j} \in V (G_{2})} {(d_{G_{1}} (u_{i}) + d_{G_{2}} (v_{j}))}^{2} . \end{array}

On the other hand, by (1)

\leq {[\frac{\sum_{u_{i} \in V (G_{1})} \sum_{v_{j} \in V (G_{2})} (d_{G_{1}} {(u_{i})}^{2} + d_{G_{2}} {(v_{j})}^{2} + 2 d_{G_{1}} (u_{i}) d_{G_{2}} (v_{j}))}{n_{1} n_{2}}]}^{n_{1} n_{2}} .

(8)

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

\leq {[\frac{\sum_{u_{i} \in V (G_{1})} (d_{G_{1}} {(u_{i})}^{2} \sum_{v_{j} \in V (G_{2})} 1 + \sum_{v_{j} \in V (G_{2})} d_{G_{2}} {(v_{j})}^{2} + 2 d_{G_{1}} (u_{i}) \sum_{v_{j} \in V (G_{2})} d_{G_{2}} (v_{j}))}{n_{1} n_{2}}]}^{n_{1} n_{2}}

which equals to

{[\frac{n_{2} M_{1} (G_{1}) + n_{1} M_{1} (G_{2}) + 8 m_{1} m_{2}}{n_{1} n_{2}}]}^{n_{1} n_{2}} .

Moreover, the equality holds in (8) if and only if $d_{G_{1}} (u_{i}) + d_{G_{2}} (v_{j}) = d_{G_{1}} (u_{k}) + d_{G_{2}} (v_{ℓ})$ for any $(u_{i}, v_{j}), (u_{k}, v_{ℓ}) \in V (G_{1} ⊠ G_{2})$ 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 $d_{G_{2}} (v_{j}) = d_{G_{2}} (v_{ℓ})$ , $v_{j}, v_{ℓ} \in V (G_{2})$ . Hence the equality holds in (6) if and only if both $G_{1}$ and $G_{2}$ are regular graphs, that is, $G_{1} ⊠ G_{2}$ is a regular graph. This completes the first part of the proof.

By the definition of the second multiplicative Zagreb index, we have

\prod_{2} (G_{1} ⊠ G_{2}) = \prod_{(u_{i}, v_{j}) (u_{k}, v_{ℓ}) \in E (G_{1} ⊠ G_{2})} (d_{G_{1}} (u_{i}) + d_{G_{2}} (v_{j})) (d_{G_{1}} (u_{k}) + d_{G_{2}} (v_{ℓ})) .

This actually can be written as

\begin{array}{rcl} \prod_{2} (G_{1} ⊠ G_{2}) & = & \prod_{u_{i} \in V (G_{1})} \prod_{v_{j} v_{ℓ} \in E (G_{2})} (d_{G_{1}} (u_{i}) + d_{G_{2}} (v_{j})) (d_{G_{1}} (u_{i}) + d_{G_{2}} (v_{ℓ})) \\ \times \prod_{v_{j} \in V (G_{2})} \prod_{u_{i} u_{k} \in E (G_{1})} (d_{G_{1}} (u_{i}) + d_{G_{2}} (v_{j})) (d_{G_{1}} (u_{k}) + d_{G_{2}} (v_{j})) \end{array}

or, equivalently,

\begin{array}{rcl} \prod_{2} (G_{1} ⊠ G_{2}) & = & \prod_{u_{i} \in V (G_{1})} \prod_{v_{j} \in V (G_{2})} {(d_{G_{1}} (u_{i}) + d_{G_{2}} (v_{j}))}^{d_{G_{2}} (v_{j})} \\ \times \prod_{v_{j} \in V (G_{2})} \prod_{u_{i} \in V (G_{1})} {(d_{G_{1}} (u_{i}) + d_{G_{2}} (v_{j}))}^{d_{G_{1}} (u_{i})} . \end{array}

After that, by (2) we get

\begin{array}{rcl} \prod_{2} (G_{1} ⊠ G_{2}) & \leq & \prod_{u_{i} \in V (G_{1})} {[\frac{\sum_{v_{j} \in V (G_{2})} d_{G_{2}} (v_{j}) (d_{G_{1}} (u_{i}) + d_{G_{2}} (v_{j}))}{2 m_{2}}]}^{2 m_{2}} \\ \times \prod_{v_{j} \in V (G_{2})} {[\frac{\sum_{u_{i} \in V (G_{1})} d_{G_{1}} (u_{i}) (d_{G_{1}} (u_{i}) + d_{G_{2}} (v_{j}))}{2 m_{1}}]}^{2 m_{1}} . \end{array}

(9)

Moreover, since

By (1) the final statement in (9) becomes

(10)

Hence the second part of the proof is over.

The equality holds in (9) and (10) if and only if $d_{G_{2}} (v_{j}) = d_{G_{2}} (v_{ℓ})$ for any $v_{j}, v_{ℓ} \in V (G_{2})$ 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, $G_{1} ⊠ G_{2}$ 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

\prod_{1} (C_{p} ⊠ K_{q}) = {(q + 1)}^{2 p q} and \prod_{2} (C_{p} ⊠ K_{q}) = {(q + 1)}^{(q + 1) p q} .

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 $E (G_{1} \circ G_{2}) = {((u_{i}, v_{0}), (u_{k}, v_{0})), (u_{i}, u_{k}) \in E (G_{1})} \cup$ ${((u_{i}, v_{j}), (u_{i}, v_{ℓ})), (v_{j}, v_{ℓ}) \in E (G_{2}), i = 1, 2, \dots, n_{1}} \cup$ ${((u_{i}, v_{0}), (u_{i}, v_{ℓ})), ℓ = 1, 2, \dots, n_{2}, i = 1, 2, \dots, n_{1}}$ . 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)
$\prod_{1} (G_{1} \circ G_{2}) \leq \frac{1}{n_{1}^{n_{1}} n_{2}^{n_{1} n_{2}}} M_{1} {(G_{1})}^{n_{1}} {(M_{1} (G_{2}) + 4 m_{2} + n_{2})}^{n_{1} n_{2}},$
(11)
(ii)
$\begin{array}{rcl} \prod_{2} (G_{1} \circ G_{2}) & \leq & {[\frac{M_{2} (G_{1}) + n_{2} M_{1} (G_{1}) + n_{2}^{2}}{m_{1}}]}^{m_{1}} {[\frac{M_{2} (G_{2}) + M_{1} (G_{2}) + 1}{m_{2}}]}^{n_{1} m_{2}} \\ \times {[\frac{4 m_{1} m_{2} + n_{1} n_{2}^{2} + 2 m_{1} n_{2} + 2 m_{2} n_{1} n_{2}}{n_{1} n_{2}}]}^{n_{1} n_{2}}, \end{array}$
(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

\begin{array}{rcl} \prod_{1} (G_{1} \circ G_{2}) & = & \prod_{(u_{i}, v_{j}) \in V (G_{1} \circ G_{2})} d_{G_{1} \circ G_{2}} {(u_{i}, v_{j})}^{2} \\ = & \prod_{u_{i} \in V (G_{1})} {(d_{G_{1}} (u_{i}) + n_{2})}^{2} \prod_{u_{i} \in V (G_{1})} \prod_{v_{j} \in V (G_{2})} {(d_{G_{2}} (v_{j}) + 1)}^{2} \\ = & \prod_{u_{i} \in V (G_{1})} (d_{G_{1}} {(u_{i})}^{2} + 2 n_{2} d_{G_{1}} (u_{i}) + n_{2}^{2}) \\ \times {[\prod_{v_{j} \in V (G_{2})} {(d_{G_{2}} {(v_{j})}^{2} + 2 d_{G_{2}} (v_{j}) + 1)}^{2}]}^{n_{1}} \\ \leq & {[\frac{\sum_{u_{i} \in V (G_{1})} (d_{G_{1}} {(u_{i})}^{2} + 2 n_{2} d_{G_{1}} (u_{i}) + n_{2}^{2})}{n_{1}}]}^{n_{1}} \\ \times {[\frac{\sum_{v_{j} \in V (G_{2})} (d_{G_{2}} {(v_{j})}^{2} + 2 d_{G_{2}} (v_{j}) + 1)}{n_{2}}]}^{n_{1} n_{2}} by (1) \\ = & \frac{1}{n_{1}^{n_{1}} n_{2}^{n_{1} n_{2}}} {(M_{1} (G_{1}) + 4 n_{2} m_{1} + n_{1} n_{2}^{2})}^{n_{1}} {(M_{1} (G_{2}) + 4 m_{2} + n_{2})}^{n_{1} n_{2}} . \end{array}

(13)

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 $d_{G_{2}} (v_{j}) = d_{G_{2}} (v_{ℓ})$ , $v_{j}, v_{ℓ} \in V (G_{2})$ , 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

\begin{array}{rcl} \prod_{2} (G_{1} \circ G_{2}) & = & \prod_{(u_{i}, v_{j}) (u_{k}, v_{ℓ}) \in E (G_{1} \circ G_{2})} d_{G_{1} \circ G_{2}} (u_{i}, v_{j}) d_{G_{1} \circ G_{2}} (u_{k}, v_{ℓ}) \\ = & \prod_{u_{i} u_{k} \in E (G_{1})} (d_{G_{1}} (u_{i}) + n_{2}) (d_{G_{1}} (u_{k}) + n_{2}) \\ \times \prod_{u_{i} \in V (G_{1})} \prod_{v_{j} \in V (G_{2})} (d_{G_{1}} (u_{i}) + n_{2}) (d_{G_{2}} (v_{j}) + 1) \\ \times \prod_{u_{i} \in V (G_{1})} \prod_{v_{j} v_{ℓ} \in E (G_{2})} (d_{G_{2}} (v_{j}) + 1) (d_{G_{2}} (v_{ℓ}) + 1) \\ = & \prod_{u_{i} u_{k} \in E (G_{1})} (d_{G_{1}} (u_{i}) d_{G_{1}} (u_{k}) + n_{2} (d_{G_{1}} (u_{i}) + d_{G_{1}} (u_{k})) + n_{2}^{2}) \\ \times {[\prod_{u_{i} \in V (G_{1})} (d_{G_{1}} (u_{i}) + n_{2})]}^{n_{2}} {[\prod_{v_{j} \in V (G_{2})} (d_{G_{1}} (v_{j}) + 1)]}^{n_{1}} \\ \times {[\prod_{v_{j} v_{ℓ} \in E (G_{2})} (d_{G_{2}} (v_{j}) d_{G_{2}} (v_{ℓ}) + (d_{G_{2}} (v_{j}) + d_{G_{2}} (v_{ℓ})) + 1)]}^{n_{1}} \\ \leq & {[\frac{M_{2} (G_{1}) + n_{2} M_{1} (G_{1}) + n_{2}^{2} m_{1}}{m_{1}}]}^{m_{1}} \times {[\frac{2 m_{1} + n_{1} n_{2}}{n_{1}}]}^{n_{1} n_{2}} \\ \times {[\frac{2 m_{2} + n_{2}}{n_{2}}]}^{n_{1} n_{2}} \times {[\frac{M_{2} (G_{2}) + M_{1} (G_{2}) + m_{2}}{m_{2}}]}^{n_{1} m_{2}} by (1) . \end{array}

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 $d_{G_{2}} (v_{j}) = d_{G_{2}} (v_{ℓ})$ for any $v_{j}, v_{ℓ} \in V (G_{2})$ , 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 $\prod_{1} (C_{p} \circ K_{q}) = q^{2 p q} {(q + 2)}^{2 p}$ 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 $(u_{k}, v_{ℓ})$ 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)
$\prod_{1} (G_{1} [G_{2}]) \leq \frac{1}{{(n_{1} n_{2})}^{n_{1} n_{2}}} {[n_{2}^{3} M_{1} (G_{1}) + 8 n_{2} m_{1} m_{2} + n_{1} M_{1} (G_{2})]}^{n_{1} n_{2}},$
(14)
(ii)
$\begin{array}{rcl} \prod_{2} (G_{1} [G_{2}]) & \leq & \frac{1}{{(n_{1} m_{2})}^{n_{1} m_{2}}} {[m_{2} n_{2}^{2} M_{1} (G_{1}) + 2 n_{2} m_{1} M_{1} (G_{2}) + n_{1} M_{2} (G_{2})]}^{n_{1} m_{2}} \\ \times \frac{1}{{(n_{2} m_{1})}^{m_{1} n_{2}^{2}}} {[n_{2}^{3} M_{2} (G_{1}) + m_{1} M_{1} (G_{2}) + 2 m_{2} n_{2} M_{1} (G_{1})]}^{n_{2}^{2} m_{1}}, \end{array}$
(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

(16)

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 $d_{G_{2}} (v_{j}) = d_{G_{2}} (v_{ℓ})$ , $v_{j}, v_{ℓ} \in V (G_{2})$ (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

(17)

(18)

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 $d_{G_{2}} (v_{j}) = d_{G_{2}} (v_{ℓ})$ , $v_{j}, v_{ℓ} \in V (G_{2})$ (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 $\prod_{1} (C_{p} [C_{q}]) = 2^{2 p q} {(q + 1)}^{2 p q}$ and $\prod_{2} (C_{p} [C_{q}]) = 2^{2 p q (q + 1)} {(q + 1)}^{2 p q (q + 1)}$ .

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 $(u_{k}, v_{ℓ})$ whenever $u_{i} u_{k} \in E (G_{1})$ or $v_{j} v_{ℓ} \in E (G_{2})$ .

Theorem 5 The first and second multiplicative Zagreb indices of the disjunction are computed as follows:

(i)
$\begin{array}{rcl} \prod_{1} (G_{1} \otimes G_{2}) & \leq & \frac{1}{{(n_{1} n_{2})}^{n_{1} n_{2}}} [n_{2}^{3} M_{1} (G_{1}) + n_{1}^{3} M_{1} (G_{2}) + M_{1} (G_{1}) M_{1} (G_{2}) \\ {+ 8 n_{1} n_{2} m_{1} m_{2} - 4 n_{1} m_{1} M_{1} (G_{2}) - 4 n_{2} m_{2} M_{1} (G_{1})]}^{n_{1} n_{2}}, \end{array}$
(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

(21)

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 $d_{G_{2}} (v_{j}) = d_{G_{2}} (v_{ℓ})$ , $v_{j}, v_{ℓ} \in V (G_{2})$ (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

\begin{array}{rcl} \prod_{2} (G_{1} \otimes G_{2}) & = & \prod_{(u_{i}, v_{j}) (u_{k}, v_{ℓ}) \in E (G_{1} \otimes G_{2})} d_{G_{1} \otimes G_{2}} (u_{i}, v_{j}) d_{G_{1} \otimes G_{2}} (u_{k}, v_{ℓ}) \\ = & \prod_{u_{i} \in V (G_{1})} \prod_{v_{j} \in V (G_{2})} P^{P}, \end{array}

where

P = n_{2} d_{G_{1}} (u_{i}) + n_{1} d_{G_{2}} (v_{j}) - d_{G_{1}} (u_{i}) d_{G_{2}} (v_{j}) .

Using the weighted arithmetic-geometric mean inequality in (2), $\prod_{2} (G_{1} \otimes G_{2})$ is less than or equal to

(22)

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}) .

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 $d_{G_{2}} (v_{j}) = d_{G_{2}} (v_{ℓ})$ , where $v_{j}, v_{ℓ} \in V (G_{2})$ (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 $\prod_{1} (K_{p} \otimes C_{q}) = {(p q - q + 2)}^{2 p q}$ and $\prod_{2} (K_{p} \otimes C_{q}) = {(p q - q + 2)}^{p q (p q - q + 2)}$ .

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 $(u_{k}, v_{ℓ})$ whenever $u_{i}$ is adjacent to $u_{k}$ in $G_{1}$ or $v_{i}$ is adjacent to $v_{ℓ}$ in $G_{2}$ , but not both. The degree of a vertex $(u_{i}, v_{j})$ of $G_{1} \oplus G_{2}$ is given by

d_{G_{1} \oplus G_{2}} (u_{i}, v_{j}) = 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}),

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)
$\begin{array}{rcl} \prod_{1} (G_{1} \oplus G_{2}) & \leq & \frac{1}{{(n_{1} n_{2})}^{n_{1} n_{2}}} [n_{2}^{3} M_{1} (G_{1}) + n_{1}^{3} M_{1} (G_{2}) + 4 M_{1} (G_{1}) M_{1} (G_{2}) \\ {+ 8 n_{1} n_{2} m_{1} m_{2} - 8 n_{1} m_{1} M_{1} (G_{2}) - 8 n_{2} m_{2} M_{1} (G_{1})]}^{n_{1} n_{2}}, \end{array}$
(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

d_{G_{1} \oplus G_{2}} (u_{i}, v_{j}) = 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}) .

By the definition of the first multiplicative Zagreb index, we have

(25)

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 $d_{G_{2}} (v_{j}) = d_{G_{2}} (v_{ℓ})$ , $v_{j}, v_{ℓ} \in V (G_{2})$ (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

\begin{array}{rcl} \prod_{2} (G_{1} \oplus G_{2}) & = & \prod_{(u_{i}, v_{j}) (u_{k}, v_{ℓ}) \in E (G_{1} \otimes G_{2})} d_{G_{1} \otimes G_{2}} (u_{i}, v_{j}) d_{G_{1} \otimes G_{2}} (u_{k}, v_{ℓ}) \\ = & \prod_{u_{i} \in V (G_{1})} \prod_{v_{j} \in V (G_{2})} P^{P}, \end{array}

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 arithmetic-geometric mean inequality in (2), we get

(26)

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 $d_{G_{2}} (v_{j}) = d_{G_{2}} (v_{ℓ})$ , $v_{j}, v_{ℓ} \in V (G_{2})$ (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 $\prod_{1} (G_{1} \oplus G_{2}) = {(p + q - 2)}^{2 p q}$ and $\prod_{2} (G_{1} \oplus G_{2}) = {(p + q - 2)}^{p q (p + q - 2)}$ .

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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 (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.

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Department of Mathematics, Sungkyunkwan University, Suwon, 440-746, Republic of Korea
Kinkar C Das
Department of Mathematics, Faculty of Arts and Science, Uludağ University, Gorukle Campus, Bursa, 16059, Turkey
Aysun Yurttas, Muge Togan & Ismail Naci Cangul
Department of Mathematics, Faculty of Science, Selçuk University, Campus, Konya, 42075, Turkey
Ahmet Sinan Cevik

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Kinkar C Das
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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/10.1186/1029-242X-2013-90

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Received: 19 December 2012
Accepted: 15 February 2013
Published: 05 March 2013
DOI: https://doi.org/10.1186/1029-242X-2013-90

The multiplicative Zagreb indices of graph operations

Abstract

1 Introduction

2 Multiplicative Zagreb index of graph operations

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