On the sharp upper and lower bounds of multiplicative Zagreb indices of graphs with connectivity at most k

# On the sharp upper and lower bounds of multiplicative Zagreb indices of graphs with connectivity at most k

Department of Mathematics and Computer Science, Adelphi University, Garden City, NY 11550, USA
###### Abstract

For a (molecular) graph, the first multiplicative Zagreb index is the product of the square of every vertex degree, and the second multiplicative Zagreb index is the product of the products of degrees of pairs of adjacent vertices. In this paper, we explore graphs in terms of (edge) connectivity. The maximum and minimum values of and of graphs with connectivity at most are provided. In addition, the corresponding extremal graphs are characterized, and our results extend and enrich some known conclusions.

Keywords: Connectivity; Edge connectivity; Extremal bounds; Multiplicative Zagreb indices.
AMS subject classification: 05C05, 05C12

## 1 Introduction

A topological index is a single number which can be used to describe some properties of a molecular graph that is a finite simple graph, representing the carbon-atom skeleton of an organic molecule of a hydrocarbon. In recent decades these could be useful for the study of quantitative structure-property relationships (QSPR) and quantitative structure-activity relationships (QSAR) and for the structural essence of biological and chemical compounds. The well-known Randi index is one of the most important topological indices.

In 1975, Randi introduced a moleculor quantity of branching index [1], which has been known as the famous Randi connectivity index that is most useful structural descriptor in QSPR and QSAR, see [2, 3, 4, 5]. Mathematicians have considerable interests in the structural and applied issues of Randi connectivity index, see [6, 7, 8, 9]. Based on the successful considerations, Zagreb indices[10] are introduced as an expected formula for the total -electron energy of conjugated molecules as follows.

 M1(G)=∑u∈V(G)d(u)2  and M2(G)=∑uv∈E(G)d(u)d(v),

where is a (molecular) graph, is a bond between two atoms and , and (or , respectively) is the number of atoms that are connected with (or , respectively). Zagreb indices are employed as molecular descriptors in QSPR and QSAR, see [11, 12]. Recently, Todeschini et al.(2010) [13, 14] proposed the following multiplicative variants of molecular structure descriptors:

 ∏1(G)=∏u∈V(G)d(u)2  and∏2(G)=∏uv∈E(G)d(u)d(v)=∏u∈V(G)d(u)d(u).

In the interdisplinary of mathemactics, chemistry and physics, it is not surprising that there are numerous studies of properties of the (multiplicative)Zagreb indices of molecular graphs [15, 16, 17, 18, 19, 20, 21].

In view of these results, researchers are intereted with finding upper and lower bounds for multiplicative Zagreb indices of graphs and characterizing the graphs in which the maximal (respectively, minimal) index values are attained, respectively. In fact, investigations of the above problems, mathematical and computational properties of Zagreb indices have also been considered in [22, 23, 24]. Other directions of investigation include studies of relation between multiplicative Zagreb indices and the corresponding invariant of elements of the graph G (vertices, pendent vertices, diameter, maximum degree, girth, cut edge, cut vertex, connectivity, perfect matching). As examples, the first and second multiplicative Zagreb indices for a class of chemical dendrimers are explored by Iranmanesh et al. [25]. Based on trees, unicyclic graphs and bicyclic graphs, Borovićanin et al. [26] introduced the bounds on Zagreb indices with a fixed domination number. The maximum and minimum Zagreb indices of trees with given number of vertices of maximum degree are proposed by Borovianin and Lampert[27]. Xu and Hua [28] introduced an unified approach to characterize extremal maximal and minimal multiplicative Zagreb indices, respectively. Considering the high dimension trees, -trees, Wang and Wei [29] provided the maximum and minimum indices of these indices and the corresponding extreme graphs are provided. Some sharp upper bounds for -index and -index in terms of graph parameters are investigated by Liu and Zhang [31], including an order, a size and a radius. Wang et al. [32] provided sharp bounds for these indices of of trees with given number of vertices of maximum degree. The bounds for the moments and the probability generating function of these indices in a randomly chosen molecular graph with tree structure of order are studied by Kazemi [33]. Li and Zhao obtained upper bounds on Zagreb indices of bicyclic graphs with a given matching number [34].

In light of the information available for multiplicative Zagreb indices, and inspired by above results, in this paper we further investigate these indices of graphs with (edge) connectivity. We give some basic properties of the first and the second multiplicative Zagreb indices. The maximum and minimum values of and of graphs with (edge) connectivity at most are provided. In addition, the corresponding extreme graphs are charaterized. In our exposition we will use the terminology and notations of (chemical) graph theory(see [35, 36]).

## 2 Preliminaries

Let be a simple connected graph, denoted by , in which is vertex set and is edge set. If a vertex , then the neighborhood of denotes the set , and (or ) is the degree of with . is the number of vertices of degree . For and , we use for the subgraph of induced by the vertex set , for the subgraph induced by and for the subgraph of G obtained by deleting . If contains at least 2 components, then is said to be a vertex cut set of . Similarly, if contains at least 2 components, then is called an edge cut set.

A graph is said to be -connected with , if either is complete graph , or it has at least vertices and contains no -vertex cut. The connectivity of , denoted by , is defined as the maximal value of for which a connected graph is -connected. Similarly, for a graph is called -edge-connected if it has at least two vertices and does not contain an -edge cut. The maximal value of for which a connected graph is -edge-connected is said to be the edge connectivity of , denoted by . According to above definitions, the following proposition is obtained.

###### Proposition 2.1.

Let be a graph with vertices. Then
(i)
(ii) and are equivalent.

Let be a set of graphs with vertices and Denote by a set of graphs with vertices and For and , is a tree. Let and be special trees: a path and a star of n vertices. is a complete graph. The graph is obtained by joining vertices of to an isolated vertex, see Fig 1. Then .

Considering the concepts of and , the following proposition is routinely obtained.

###### Proposition 2.2.

Let be an edge of a graph (, respectively). Then
(i) (, respectively),
(ii)

In addition, by elementary calculations, these two statements are deduced.

###### Proposition 2.3.

If , then is an increasing function.

###### Proposition 2.4.

If , then is a decreasing function.

## 3 Lemmas and main results

In this section, the maximal and minimal multiplicative Zagreb indices of graphs with connectivity at most in and are determined. The corresponding extremal graphs shall be characterized. We first provide some lemmas, which are very important and will be used in the proof of our main results.

###### Lemma 3.1.

[25] Let be a tree of n vertices. If is not or , then and .

Considering the definitions of and , we have the following lemma.

###### Lemma 3.2.

Let and . Then

 ∏1(G+uv)>∏1(G) and ∏2(G+uv)>∏1(G).

Given two graphs and , if , then the join graph is a graph with vertex set and edge set .

###### Lemma 3.3.

Let be a graph with vertices, in which and are cliques, and is a graph with vertices, see Fig 2. If and , then

 ∏1(G(j,n−k−j))<∏1(G(1,n−k−1)).
###### Proof.

We consider the graph from to . Note that if in , then ; if in , then ; if in , then . By the concepts of and , we have

 ∏1(G1)∏1(G2) =∏v∈V(Kj)d(v)2∏v∈V(Hk)d(v)2∏v∈V(Kn−k−j)d(v)2∏v∈V(Kj−1)d(v)2∏v∈V(Hk)d(v)2∏v∈V(Kn−k−j+1)d(v)2 =((j+k−1)2)j((n−j−1)2)n−k−j((j+k−2)2)j−1((n−j)2)n−k−j+1 =((j+(k−1))j((j−1)+(k−1))j−1((n−j−k+1)+(k−1))n−k−j+1((n−j−k)+(k−1))n−k−j)2.

Since , then . By Proposition 2.3 and , we have

 ∏1(G1)∏1(G2)<1,

that is, .

We can recursively use this process from to , and obtain that

 ∏1(G(j,n−k−j))<∏1(G(j−1,n−k−j+1))<∏1(G(j−2,n−k−j+2))<⋯<∏1(G(1,n−k−1)).

Therefore, . Thus, we complete the proof. ∎

###### Lemma 3.4.

Let be a connected graph and . Assume that , . Let . If and is not adjacent to , then

 ∏2(G′)>∏2(G).
###### Proof.

By the concept of , we have

 ∏2(G)∏2(G′)=d(u)d(u)d(v)d(v)(d(u)+s)d(u)+s(d(v)−s)d(v)−s=(d(u)d(u)(d(u)+s)d(u)+s)((d(v)−s)d(v)−sd(v)d(v)).

Since and by Proposition 2.4, then

 ∏2(G)∏2(G′)<1,

that is, . Thus, the lemma is proved. ∎

###### Lemma 3.5.

If and , we have

 ∏2(G(j,n−k−j))<∏2(G(1,n−k−1)).
###### Proof.

Let and . Note that vertex set . We create a new graph . By and Lemma 3.4, we have .

Note that for , has neighbors in only. Let and . By Lemma 3.2, we have . Therefore and . Thus the proof is complete. ∎

Next we will turn to prove our main results. In Theorems 3.1 and 3.2, the sharp upper bounds of mutiplicative Zagreb indices of graphs in and are proposed.

###### Theorem 3.1.

Let be a graph in . Then

 ∏1(G)≤k2(n−k)2k(n−2)2(n−k−1) and ∏2(G)≤kk(n−1)k(n−1)(n−2)(n−2)(n−k−1),

where the equalities hold if and only if .

###### Proof.

Note that the degree sequence of is . By the concepts of , and routine calculations, we have

 ∏1(Kkn)=k2(n−1)2k(n−2)2(n−k−1) and ∏2(Kkn)=kk(n−1)k(n−1)(n−2)(n−2)(n−k−1).

It suffices to prove that and , and the equalities hold if and only if .

If , then , and the theorem is true. If , then choose a graph (, respectively) in such that (, respectively) is maximal. Since with , then has a vertex cut set of size . Let be the cut vertex set of . Denoted by the number of components of . In order to prove our theorem, we start with several claims.

###### Claim 1.

with .

Proof. We proceed to prove it by a contradiction. Assume that with . Let be the components of . Since , then choose vertices and . Then is still a -vertex cut set of . By Lemma 3.2, we have , a contradiction to the choice of . Thus, this claim is proved.

Without loss of generality, suppose that contains only two connected components, denoted by and .

###### Claim 2.

The induced subgraphs of and in are complete subgraphs.

Proof. We use a contradiction to show it. Suppose that is not a complete subgraph of . Then there exists an edge . Since , by Lemma 3.2, we have , which is a contadiction. Thus, we show this claim.

By the above claims, we see that and are complete subgraph of . Let and . Then we have .

###### Claim 3.

Either or .

Proof. On the contrary, assume that . Without loss of generility, For , by Lemmas 3.3 and 3.5, we have a new graph such that and . This is a contradition to the choice of . Thus, either or , and this claim is showed.

By Lemma 3.2, . Since , then is maximal and this theorem holds. ∎

Since , then the following result is immediate.

###### Theorem 3.2.

Let be a graph in . Then

 ∏1(G)≤k2(n−k)2k(n−2)2(n−k−1)and ∏2(G)≤kk(n−1)k(n−1)(n−2)(n−2)(n−k−1),

where the equalities hold if and only if .

In the rest of this paper, we consider the minimal mutiplicative Zagreb indices of graphs in and . By Proposition 2.2 (ii), is a tree with vertices. By Lemma 3.1 and routine calculations, we have

###### Theorem 3.3.

Let be a graph in . Then

 ∏1(G)≥(n−1)2 and ∏2(G)≥4n−2,

where the equalities hold if and only if and , respectively.

Note that , then the following theorem is obvious.

###### Theorem 3.4.

Let be a graph in . Then

 ∏1(G)≥(n−1)2 and ∏2(G)≥4n−2,

where the equalities hold if and only if and , respectively.

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