On the maximum and minimum multiplicative Zagreb indices of graphs with given number of cut edges

TL;DR

This paper investigates the extremal values of the first and second multiplicative Zagreb indices in graphs with a fixed number of cut edges, providing characterizations and extending known results.

Contribution

It characterizes graphs with maximum and minimum Zagreb indices among those with specified vertices and cut edges, extending previous findings.

Findings

Identified graphs with extremal Zagreb indices for given cut edges.

Provided formulas and characterizations for maximum and minimum indices.

Extended known results in graph theory related to Zagreb indices.

Abstract

For a molecular graph, the first multiplicative Zagreb index $\Pi_1$ is equal to the product of the square of the degree of the vertices, while the second multiplicative Zagreb index $\Pi_2$ is equal to the product of the endvertex degree of each edge over all edges. Denote by $\mathbb{G}_{n,k}$ the set of graphs with $n$ vertices and $k$ cut edges. In this paper, we explore graphs in terms of a number of cut edges. In addition, the maximum and minimum multiplicative Zagreb indices of graphs with given number of cut edges are provided. Furthermore, we characterize graphs with the largest and smallest $\Pi_1(G)$ and $\Pi_2(G)$ in $\mathbb{G}_{n,k}$, and our results extend and enrich some known conclusions.

Equations

$$M_1(G)=\sum _{u\in V(G)}d(u{)}^2\text{ and}{M}_2(G)=\sum _{uv\in E(G)}d(u)d(v).$$

$${\Pi }_1(G)=\prod _{u\in V(G)}d(u{)}^2\text{ and}{\Pi }_2(G)=\prod _{uv\in E(G)}d(u)d(v)=\prod _{u\in V(G)}d(u{)}^{d(u)}.$$

$$\displaystyle\frac{\Pi_{1}(G_{1}P_{m}G_{2})}{\Pi_{1}(G_{1}G_{2}P_{m})}=\frac{x^{2}y^{2}}{(x+y-1)^{2}1^{2}}=\bigg{(}\frac{\frac{x}{x+y-1}}{\frac{1}{1+(y-1)}}\bigg{)}^{2}.$$

$$\frac{{\Pi }_2(G_1{P}_m{G}_2)}{{\Pi }_2(G_1{G}_2{P}_m)}=\frac{x^x{y}^y}{(x+y-1{)}^{(x+y-1)}{1}^1}=\frac{\frac{x^x}{{(x+y-1)}^{(x+y-1)}}}{\frac{1^1}{{(1+y-1)}^{(1+y-1)}}}.$$

$$\frac{{\Pi }_1(G)}{{\Pi }_1(G^{\prime })}=\frac{x^2{y}^2}{(x-s{)}^2(y+s{)}^2}=\frac{(\frac{y}{y+s}{)}^2}{(\frac{x-s}{(x-s)+s}{)}^2},$$

$$\frac{{\Pi }_2(G)}{{\Pi }_2(G^{\prime })}=\frac{x^x{y}^y}{(x-s{)}^{x-s}(y+s{)}^{y+s}}=\frac{\frac{y^y}{(y+s{)}^{y+s}}}{\frac{(x-s{)}^{x-s}}{x^x}}=\frac{\frac{y^y}{(y+s{)}^{y+s}}}{\frac{(x-s{)}^{x-s}}{{[(x-s)+s]}^{(x-s)+s}}},$$

$$\frac{{\Pi }_1(G)}{{\Pi }_1(G^{\prime \prime })}=\frac{x^2{y}^2}{(x+t{)}^2(y-t{)}^2}=\frac{(\frac{x}{x+t}{)}^2}{(\frac{y-t}{(y-t)+t}{)}^2}$$

$$\frac{{\Pi }_2(G)}{{\Pi }_2(G^{\prime \prime })}=\frac{x^x{y}^y}{(x+t{)}^{x+t}(y-t{)}^{y-t}}=\frac{\frac{x^x}{(x+t{)}^{x+t}}}{\frac{(y-t{)}^{y-t}}{y^y}}=\frac{\frac{x^x}{(x+t{)}^{x+t}}}{\frac{(y-t{)}^{y-t}}{[(y-t)+t{]}^{(y-t)+t}}}.$$

$$\displaystyle\frac{\Pi_{1}(G)}{\Pi_{1}(G^{\prime})}=\frac{d(u_{s})^{2}d(v_{1})^{2}}{d_{G^{\prime}}(u_{s})^{2}d_{G^{\prime}}(v_{1})^{2}}=\bigg{(}\frac{\frac{1}{2}}{\frac{d(v_{1})-1}{d(v_{1})}}\bigg{)}^{2}.$$

$$\displaystyle\frac{\Pi_{2}(G)}{\Pi_{2}(G^{\prime})}=\frac{d(u_{s})^{d(u_{s})}d(v_{1})^{d(v_{1})}}{d_{G^{\prime}}(u_{s})^{d_{G^{\prime}}(u_{s})}d_{G^{\prime}}(v_{1})^{d_{G^{\prime}}(v_{1})}}=\bigg{(}\frac{\frac{~{}1^{1}~{}}{~{}2^{2}~{}}}{\frac{(d(v_{1})-1)^{d(v_{1})-1}}{d(v_{1})^{d(v_{1})}}}\bigg{)}^{2}.$$

$${\Pi }_1(G)\ge 4^{n-k-1}(k+2{)}^2,$$

$${\Pi }_2(G)\ge 27\ast 4^{n-2},$$

$${\Pi }_1(G)\le 4^{k-1}(n-k{)}^2(n-k-1{)}^{2(n-k-1)},$$

$$\frac{{\Pi }_1(G)}{{\Pi }_1(G^{\prime })}$$

$$\frac{{\Pi }_1(G_1)}{{\Pi }_1(G_2)}\le \frac{3n_2(3-1{)}^{3-1}(n_2-1{)}^{n_2-1}}{(3+n_2-2{)}^{3+n_2-1}}=\frac{12\ast n_2\ast (n_2-1{)}^{n_2-1}}{(n_2+1{)}^2(n_2+1)(n_2+1{)}^{n_2-1}}.$$

$$\frac{\Pi_{1}(G)}{\Pi_{1}(G^{\prime\prime})}=\frac{d(u_{1})^{2}d(u_{s})^{2}}{d_{G^{\prime\prime}}(u_{1})^{2}d_{G^{\prime\prime}}(u_{s})^{2}}=\bigg{(}\frac{\frac{~{}1~{}}{2}}{\frac{d(u_{1})-1}{d(u_{1})}}\bigg{)}^{2}.$$

$${\Pi }_2(G)\le (n-1{)}^{n-1}(n-k-1{)}^{(n-k-1{)}^2},$$

Full text

Keywords: Cut edge; Graph transformation; Multiplicative Zagreb indices; Extremal values.

AMS subject classification: 05C12, 05C35

1 Introduction

In the interdisciplinary of mathematics, chemistry and physics, molecular invariants/descriptors can be applied the study of quantitative structure-property relationships (QSPR) and quantitative structure-activity relationships (QSAR), and for the descriptive purposes of biological and chemical properties, such as boiling and melting points and toxicity [1]. One type of the most classical topological molecular descriptors is named as Zagreb indices $M_{1}$ and $M_{2}$ [2], which are literal quantities in an expected formula for the total $\pi$-electron energy of conjugated molecules. In the view of successful considerations on the applications on Zagreb indices [3], Todeschini et al.(2010) [4, 5, 6] introduced the multiplicative variants of molecular structure descriptors, denoted by $\Pi_{1}$ and $\Pi_{2}$ the multiplicative Zagreb indices. (Multiplicative) Zagreb indices are employed as molecular descriptors in QSPR and QSAR, see [7, 8].

Mathematicians have exhibited considerable interest in the properties of Zagreb indices about the extremal values or bounds for the topological indices of graphs, as well as related problems of characterizing the extremal graphs [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. In addition to a plenty of applications for the Zagreb indices in chemistry, there are many situations in multiplicative Zagreb indices, which attracted one of the focus of interests in physics and graph theory. Borovićanin et al. [21] investigated upper bounds on Zagreb indices of trees in terms of domination number and extremal trees are characterized. Wang and Wei [6] introduced sharp upper and lower bounds of these indices in $k$-trees. Liu and Zhang [14] provided several sharp upper bounds for $\pi_{1}$-index and $\pi_{2}$-index in terms of graph parameters such as the order, size and radius [27]. Wang et al. [18] obtained extremal multiplicative Zagreb indices of trees with given number of vertices of maximum degree. Xu and Hua [25] explored an unified approach to characterize extremal (maximal and minimal) trees, unicyclic graphs and bicyclic graphs with respect to multiplicative Zagreb indices, respectively. Iranmanesh et al. [26] gave the first and the second multiplicative Zagreb indices for a class of chemical moleculor of dendrimers. Also, a lower bound for the first Zagreb index of trees with a given domination number is determined and the extremal trees are characterized as well. Kazemi [28] studied the bounds for the moments and the probability generating function of these indices in a randomly chosen molecular graph with tree structure of order $n$. The connected graphs with $k$ cut edges (or vertices) have been considered in many mathematical literatures [20, 22, 23, 24, 29]. It is natural to consider that, for the $n$-vertex tree, $k=n-1$, and trees with the extremal multiplicative Zagreb indices had been studied a long time ago [26].

In view of the above results, in this paper we further investigate multiplicative Zagreb indices of graphs with a given number of cut edges. In addition, the maximum and minimum of $\Pi_{1}(G)$ and $\Pi_{2}(G)$ of graphs with given number of cut edges are provided. Furthermore, we characterize graphs with the largest and smallest multiplicative Zagreb indices in $\mathbb{G}_{n,k}$.

2 Preliminary

Denote by $G=(V,E)$ a simple connected graph with $n$ vertices and $m$ edges, where $V=V(G)$ is called vertex set and $E=E(G)$ is called edge set. For $v\in V(G)$, $N(v)$ denotes the neighbors of $v$, that is, $N_{G}(v)=\left\{u|\;uv\in E(G)\right\}$, and $d_{G}(v)=\left|N(v)\right|$ is the degree of $v$. The first and second Zagreb indices [3] of a graph $G$ are given by

[TABLE]

The first multiplicative Zagreb index $\Pi_{1}=\Pi_{1}(G)$ and the second multiplicative Zagreb index $\Pi_{2}=\Pi_{2}(G)$ [4, 5] of a graph $G$ are defined as

[TABLE]

A pendent vertex is a vertex of degree one and a supporting vertex is a vertex adjacent to at least one pendent vertex. A pendent edge is incident to a pendent vertex and a supporting vertex. For two graphs $G_{1}$ and $G_{2}$, if there exists a common vertex $v$ between them, then let $G_{1}vG_{2}$ be a graph such that the vertex set of $G_{1}vG_{2}$ is $V(G_{1})\bigcup V(G_{2})$, $V(G_{1})\bigcap V(G_{2})={v}$ and $E(G_{1}vG_{2})=E(G_{1})\bigcup E(G_{2})$. If $G_{1}$, $G_{2}$, $\cdots$, $G_{l}$ with $l\geq 2$ share a common vertex $v$, then by $G_{1}vG_{2}v\cdots vG_{l}$ denote this graph. For $u_{1}\in V(G_{1})$ and $u_{s}\in V(G_{2})$, if $P=u_{1}u_{2}\cdots u_{s}$ is a path, then denote this graph by $G_{1}PG_{2}$ or $G_{1}u_{1}u_{2}\cdots u_{s}G_{2}$ in which $P$ is called an internal path. For $S\subseteq V(G)$ and $F\subseteq E(G)$, we use $G[S]$ for the subgraph of $G$ induced by $S$, $G-S$ for the subgraph induced by $V(G)-S$ and $G-F$ for the subgraph of G obtained by deleting $F$. A vertex $u$ (or an edge $e$, respectively) is called a cut vetex (or cut edge, respectively) of a connected graph $G$, if $G-v$ (or $G-e$) has at least two components. A graph G is said to be $2$-connected if there does not exist a vertex whose removal disconnects the graph. A block is a connected graph which does not have any cut vertex, and $K_{2}$ is a trivial block. The endblock contains at most one cut vertex. As usual, $P_{n}$, $S_{n}$ and $C_{n}$ are a path, a star and a cycle on $n$ vertices, respectively. The cyclomatic number of a connected graph $G$ is given by $c(G)=m-n+1$. In particular, if $c(G)=0,\,1$ and $2$, then $G$ is a tree, unicyclic graph and bicyclic graph, respectively. If a connected graph on $n$ vertices has the cyclomatic number at least one, then the number of its cut edges is at most $n-3$. Thus, we assume that $G$ has $1\leq k\leq n-3$ cut edges in our following discussion.

Let $\mathbb{G}_{n,k}$ be a set of graphs with $n$ vertices and $1\leq k\leq n-3$ cut edges, and $E_{c}=\left\{e_{1},e_{2},\cdots,e_{k}\right\}$ be a set of cut edges of $G$. Then $E_{c}$ can be considered as two categories, which are the pendent edges and non-pendent edges (or internal paths of length $1$). The components of $G-E_{c}$ are $2$-connected graphs and isolated vertices. Denote by $K_{n}^{S}$ (or $K_{n}^{P}$, respectively) a graph obtained by identifying (connecting to, respectively) the nonpendent vertex of a star $S_{k}$ (or a pendent vertex of a path $P_{k}$, respectively) to a vertex of $K_{n-k}$ (see Fig 1). In addition, let $C_{n}^{S}$ (or $C_{n}^{P}$, respectively) be a graph obtained by identifying (connecting to, respectively) the nonpendent vertex of a star $S_{k}$ (or a pendent vertex of a path $P_{k}$, respectively) to a vertex of $C_{n-k}$.

In our exposition we will use the terminology and notations of (chemical) graph theory(see [30, 31]). By elementary calculations, one can derive the following propositions.

Proposition 2.1**.**

Let $t(x)=\frac{x}{x+m}$ be a function with $m>0$. Then $t(x)$ is increasing in $\mathbb{R}$.

Proposition 2.2**.**

Let $l(x)=\frac{x^{x}}{(x+m)^{x+m}}$ be a function with $m>0$. Then $l(x)$ is decreasing in $\mathbb{R}$.

Based on the concepts of $\Pi_{1}(G)$ and $\Pi_{2}(G)$, we have

Lemma 2.1**.**

*Let $G=(V,E)$ is a graph and $i=1,\,2$.

(i) If $e=uv\notin E(G)$, $u,v\in V(G)$, then $\Pi_{i}(G+uv)>\Pi_{i}(G)$.

(ii) If $e=uv\in E(G)$, then $\Pi_{i}(G-e)<\Pi_{i}(G)$.*

Lemma 2.1 yields the following result.

Lemma 2.2**.**

*Let $G=(V,E)$ be a $2$-connected graph and $i=1,2$.

(i) If $\Pi_{i}(G)$ is maximal, then $G$ is the complete graph $K_{n}$.

(ii) If $\Pi_{i}(G)$ is minimal, then $G$ is the cycle $C_{n}$.*

Lemma 2.3**.**

Let $C_{1},C_{2}$ be cycles, and $P_{s}=u_{1}u_{2}\cdots u_{s}$ be an internal path of $G=C_{1}P_{s}C_{2}$ such that $u_{1}\in V(C_{1})$ and $u_{s}\in V(C_{2})$. Assume that $u_{1}v_{1},u_{1}v_{2}\in E(C_{1})$ and $u_{s}w_{1},u_{s}w_{2}\in E(C_{2})$. Let $G^{\prime}=G-\{u_{1}v_{2},u_{s}w_{1},u_{s}w_{2}\}+\{v_{2}w_{2},u_{1}w_{1}\}$. Then $\Pi_{i}(G)>\Pi_{i}(G^{\prime})$ with $i=1,2$.

Proof.

By the transformation from $G$ to $G^{\prime}$, we have $d_{G^{\prime}}(u_{s})=1<d_{G}(u_{s})=3$. For $v\in V(G)-\{u_{s}\}$, $d_{G}(v)=d_{G^{\prime}}(v)$. Then $\Pi_{i}(G)>\Pi_{i}(G^{\prime})$ with $i=1,2$, and we complete the proof. ∎

Lemma 2.4**.**

Let $G_{1}P_{m}G_{2}$ and $G_{1}G_{2}P_{m}$ be graphs (see Fig 2), in which $P_{m}$ is a path, and $G_{1}$, $G_{2}$ are connected. Then $\Pi_{1}(G_{1}P_{m}G_{2})\geq\Pi_{1}(G_{1}G_{2}P_{m})$ and $\Pi_{2}(G_{1}P_{m}G_{2})\leq\Pi_{2}(G_{1}G_{2}P_{m})$.

Proof.

Let $d_{G_{1}P_{m}G_{2}}(u)=x$ and $d_{G_{1}P_{m}G_{2}}(v)=y$. Then $d_{G_{1}G_{2}P_{m}}(u)=x+y-1$. By the definitions of multiplicative Zagreb indices, we obtain

[TABLE]

Since $x\geq 1,y\geq 1$, and by Propostion 2.1, we have $\Pi_{1}(G_{1}P_{m}G_{2})\geq\Pi_{1}(G_{1}G_{2}P_{m})$. Note that

[TABLE]

By $x\geq 1$ and Proposition 2.2, we have $\frac{\Pi_{2}(G_{1}P_{m}G_{2})}{\Pi_{2}(G_{1}G_{2}P_{m})}\leq 1$, that is, $\Pi_{2}(G_{1}P_{m}G_{2})\leq\Pi_{2}(G_{1}G_{2}P_{m})$. Thus, this completes the proof. ∎

From Lemma 2.4, we have the useful lemma below.

Lemma 2.5**.**

Let $GT$ be a graph by indentifying a vertex of a tree $T\ncong S_{n}$ to a vertex $u$ of $G$, and $GS$ be a graph by attaching $|E(T)|$ pendent edges to $u$ (see Fig 3). Then $\Pi_{1}(GT)>\Pi_{1}(GS)$ and $\Pi_{2}(GT)<\Pi_{2}(GS)$.

Lemma 2.6**.**

Let $u$ ($v$, respectively) be a vertex in $G$, and $u_{1},u_{2},\dots,u_{s}$ be the endvertices of pendent path $P_{1},P_{2},\cdots,P_{s}$ ($v_{1},v_{2},\dots,v_{t}$ be the endvertices of $P_{1}^{\prime},P_{2}^{\prime},\cdots,P_{t}^{\prime}$, respectively). Set $uu_{i}^{\prime}\in E(P_{i})$ with $1\leq i\leq s$, and $vv_{j}^{\prime}\in E(P_{j}^{\prime})$ with $1\leq j\leq t$. Let $G^{\prime}=G-\left\{uu_{i}^{\prime}\right\}+\left\{vu_{i}^{\prime}\right\}$ with $1\leq i\leq s$, $G^{\prime\prime}=G-\{vv_{j}^{\prime}\}+\{uv_{j}^{\prime}\}$ with $1\leq j\leq t$ and $\left|V(G_{0})\right|\geq 3$ (see Fig 4). Then either $\Pi_{1}(G)\geq\Pi_{1}(G^{\prime})$ and $\Pi_{2}(G)\leq\Pi_{2}(G^{\prime})$, or $\Pi_{1}(G)>\Pi_{1}(G^{{}^{\prime\prime}})$ and $\Pi_{2}(G)<\Pi_{2}(G^{{}^{\prime\prime}})$.

Proof.

Let $d_{G}(u)=x$, $d_{G}(v)=y$. By the constructions of $G^{\prime}$ and $G^{\prime\prime}$ , we have $d_{G^{\prime}}(u)=d_{G}(u)-s=x-s$, $d_{G^{\prime}}(v)=d_{G}(v)+s=y+s$, $d_{G^{\prime\prime}}(u)=d_{G}(u)+t=x+t$ and $d_{G^{\prime\prime}}(v)=d_{G}(v)-t=y-t.$ Combining with the concepts of mutiplicative Zagreb indices, we have

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

If $y\geq x-s$, by Propositions 2.1 and 2.2, we can obtain that $\Pi_{1}(G)\geq\Pi_{1}(G^{\prime})$ and $\Pi_{2}(G)\leq\Pi_{2}(G^{\prime})$. If $y\leq x-s-1$, then $x\geq y+s+1>y-t$. Propositions 2.1 and 2.2 yield that $\Pi_{1}(G)>\Pi_{1}(G^{\prime\prime})$ and $\Pi_{2}(G)<\Pi_{2}(G^{\prime\prime})$. Thus, the lemma is proved. ∎

Lemma 2.7**.**

Let $P_{1}=u_{1}u_{2}\cdots u_{s}$ and $P_{2}=v_{1}v_{2}\cdots v_{t}$ be two pendent path of $G$ with $s,t\geq 2$ and $d(u_{s})=d(v_{t})=1$ (see Fig 5). Let $G^{\prime}=G-v_{1}v_{2}+u_{s}v_{2}$. Then $\Pi_{1}(G)<\Pi_{1}(G^{\prime})$ and $\Pi_{2}(G)>\Pi_{2}(G^{\prime})$.

Proof.

Note that $d(u_{1})\geq 3,d(v_{1})\geq 3$. By the definitions of mutiplicative Zagreb indices, we have

[TABLE]

By Proposition 2.1, we have $\frac{\Pi_{1}(G)}{\Pi_{1}(G^{\prime})}<1$, that is, $\Pi_{1}(G)<\Pi_{1}(G^{\prime})$.

[TABLE]

By Proposition 2.2, we have $\frac{\Pi_{2}(G)}{\Pi_{2}(G^{\prime})}>1$, that is, $\Pi_{2}(G)>\Pi_{2}(G^{\prime})$.

Thus, this completes the proof. ∎

3 Graphs with smallest multiplicative Zagreb indices in $\mathbb{G}_{n,k}$

In this section, we discuss graphs with the smallest $\Pi_{1}(G)$ and $\Pi_{2}(G)$ in $\mathbb{G}_{n,k}$.

Theorem 3.1**.**

Let $G$ be a graph in $\mathbb{G}_{n,k}$ with $1\leq k\leq n-3$. Then

[TABLE]

where the equality holds if and only $G\cong C_{n}^{S}$, respectively.

Proof.

Choose a graph $G\in\mathbb{G}_{n,k}$ such that $\Pi_{1}(G)$ is as small as possible. Let $E_{c}$ be a cut edge set of $G$ and $B_{1},B_{2},\cdots,B_{k+1}$ be the components of $G-E_{c}$. By Lemma 2.2, we have $B_{i}$ is a cycle or an isolated vertex. Lemma 2.3 implies that $G$ has a unique cycle. By Lemma 2.5, all cut edges in $G$ are pendent edge. By Lemma 2.6, all pendent edges share a common supporting vertex, that is, $G\cong C_{n}^{S}$. Thus, this completes the proof. ∎

Theorem 3.2**.**

Let $G$ be a graph in $\mathbb{G}_{n,k}$ with $1\leq k\leq n-3$. Then

[TABLE]

where the equality holds if and only $G\cong C_{n}^{P}$.

Proof.

Let $G\in\mathbb{G}_{n,k}$ be a graph such that $\Pi_{2}(G)$ is minimal. Let $E_{c}$ be a cut edge set of $G$ and $B_{1},B_{2},\cdots,B_{k+1}$ be the components of $G-E_{c}$. By Lemma 2.2, we have $B_{i}$ is a cycle or an isolated vertex. Lemma 2.3 implies that $G$ has a unique cycle. By Lemma 2.7, there is only one pendent path in $G$. Thus $G\cong C_{n}^{P}$, and we prove this theorem. ∎

4 Graphs with largest multiplicative Zagreb indices in $\mathbb{G}_{n,k}$

We proceed to consider graphs with the largest $\Pi_{1}(G)$ and $\Pi_{2}(G)$ in $\mathbb{G}_{n,k}$ in this section.

Theorem 4.1**.**

Let $G$ be a graph in $\mathbb{G}_{n,k}$ with $1\leq k\leq n-3$. Then

[TABLE]

where the equality holds if and only $G\cong K_{n}^{P}$.

Proof.

Denote by a graph $G\in\mathbb{G}_{n,k}$ such that $\Pi_{1}(G)$ is maximal. Set $E_{c}$ to be a cut edge set of $G$ and $B_{1},B_{2},\cdots,B_{k+1}$ the components of $G-E_{c}$. By Lemma 2.2, we have $B_{i}$ is a clique of size at least 3 or an isolated vertex. Next we start with the follwing claims.

Claim 1**.**

Every two cliques of size at least 3 do not share a common vertex.

Proof of Claim 1. We proceed it by a contradiction. Assume there are at least two blocks $B_{1},B_{2}$ shared a common vertex $v_{0}$ in $G$ such that $|B_{1}|,|B_{2}|\geq 3$. Choose $v_{1}\in V(B_{1})$, $v_{2}\in V(B_{2})$ and $v_{1},v_{2}\neq v_{0}$. Let $G^{\prime}=G+v_{1}v_{2}$. By Lemma 2.1, $\Pi_{2}(G^{\prime})>\Pi_{2}(G)$, that is a contradiction to the assumption of $G$. The claim is proved.

We introduce a graph transformation that is used in the rest of our proof.

Claim 2**.**

Let $K_{n_{1}}$ and $K_{n_{2}}$ be two farthest endblocks of $K_{n_{1}}G_{0}K_{n_{2}}$ such that $v_{11}\in V(K_{n1})\cap V(G_{0})$ and $v_{l1}\in V(K_{n2})\cap V(G_{0})$ (see Fig 6). If $d(v_{11})=n_{1}\geq 3$ and $d(v_{l1})=n_{2}\geq 3$, then $\Pi_{1}(K_{n_{1}}G_{0}K_{n_{2}})<\Pi_{1}(K_{n_{1}+n_{2}-1}G_{0}).$

Proof of Claim 2. Let $V(K_{n_{1}})=\{v_{11},v_{12},\cdots,v_{1n_{1}}\}$ and $V(K_{n_{2}})=\{v_{l1},v_{l2},\cdots,v_{ln_{2}}\}$. Denote by $G=K_{n_{1}}G_{0}K_{n_{2}}$ and $G^{\prime}=G-\{v_{l1}v_{li},i\geq 2\}+\{v_{li}v_{1j},i\geq 2,j\geq 1\}=K_{n_{1}+n_{2}-1}G_{0}$. By the concepts of multiplicative Zagreb indices, we have

[TABLE]

Let $f(x)=\frac{xn_{2}(x-1)^{x-1}(n_{2}-1)^{n_{2}-1}}{(x+n_{2}-2)^{x+n_{2}-1}}$. Then we take a derivative of $ln({f(x)})$ as $\frac{1}{x}+ln(x-1)+1-ln(x+n_{2}-2)-\frac{x+n_{2}-1}{x+n_{2}-2}<\frac{1}{x}+ln(x-1)-ln(x+n_{2}-2)\leq\frac{1}{x}+ln(x-1)-ln(x+1)$, by $n_{2}\geq 3$.

Set $g(x)=\frac{1}{x}+ln(x-1)-ln(x+1)$. Note that $g^{\prime}(x)=\frac{x^{2}+1}{x^{2}(x^{2}-1)}>0$ and $lim_{x\rightarrow\infty}g(x)=lim_{x\rightarrow\infty}ln(\frac{(x-1)e^{\frac{1}{x}}}{x+1})=0$, by L’ Hospital’s Rule. Thus, $g(x)<0$. So, $f(x)$ is a decreasing function and

[TABLE]

Since $12\leq(n_{2}+1)^{2}$ and $n_{2}<n_{2}+1$, then $\frac{\Pi_{1}(G_{1})}{\Pi_{1}(G_{2})}<1.$ This completes the proof of Claim 2

Claim 3**.**

There exsits exactly one path.

Proof of Claim 3. We prove it by contradictions. Assume that there are at least two paths $P_{1}=u_{1}u_{2}\cdots u_{s},P_{2}=v_{1}v_{2}\cdots v_{l}$ with $d(u_{1}),d(v_{1})\geq 3$. We consider three cases that $P_{i}$ is either a pendent path or an internal path with $i=1,2$.

Case 1. $d(u_{s})=d(v_{l})=1.$

Proof of Case 1. By Lemma 2.7, there is another graph $G^{\prime}\in\mathbb{G}_{n}^{k}$ such that $\Pi_{1}(G)<\Pi_{1}(G^{\prime})$, which is a contradiction to the choice of $G$.

Case 2. $d(u_{s})=1,d(v_{l})\geq 3.$

Proof of Case 2. Let $G^{\prime\prime}=G-\{v_{1}v_{2},u_{1}u_{2}\}+\{v_{1}u_{2},v_{2}u_{s}\}$. Note that

[TABLE]

Since $d(u_{1})\geq 3$, by Proposition 2.1, we have $\Pi_{1}(G)<\Pi_{1}(G^{\prime\prime})$, that is a contradiction to the choice of $G$.

Case 3. $d(u_{s})\geq 3,d(v_{l})\geq 3.$

Proof of Case 3. By Case 2, there does not exist any pendent paths in $G$. Then every cut edge is in an internal path. By choosing two farthest endblocks and Claim 2, there is another graph $G^{\prime\prime\prime}$ such that $\Pi_{1}(G^{\prime\prime\prime})>\Pi_{1}(G)$, which contradicted that $\Pi_{1}(G)$ is maximal. This completes the proof of Case 3.

Therefore, $G$ contains a unique clique of size at least $3$ and the unique path is a pendent path. Thus $G\cong K_{n}^{P}$, and this completes the proof. ∎

Theorem 4.2**.**

Let $G$ be a graph in $\mathbb{G}_{n,k}$ with $1\leq k\leq n-3$. Then

[TABLE]

where the equality holds if and only $G\cong K_{n}^{S}$.

Proof.

Pick a graph $G\in\mathbb{G}_{n,k}$ such that $\Pi_{2}(G)$ is as large as possible. Denote by $E_{c}$ a cut edge set of $G$ and $B_{1},B_{2},\cdots,B_{k+1}$ be the components of $G-E_{c}$. By Lemma 2.2, we have $B_{i}$ is a clique of size at least 3 or an isolated vertex. By Lemma 2.4, if two blocks are connected by a path, then they share a common vertex.

Claim 4**.**

There is a uniqe block $B$ such that $|B|\geq 3$.

Proof of Claim 4. We proceed it by a contradiction. Assume that there are at least two blocks $B_{1},B_{2}$ shared a common vertex $v_{0}$ in $G$ such that $|B_{1}|,|B_{2}|\geq 3$. Choose $v_{1}\in V(B_{1})$ and $v_{2}\in V(B_{2})$ and $v_{1},v_{2}\neq v_{0}$. Let $G^{\prime}=G+v_{1}v_{2}$. By Lemma 2.1, $\Pi_{2}(G^{\prime})>\Pi_{2}(G)$ and this claim is proved.

By Lemmas 2.5 and 2.6, we have $G\cong K_{n}^{S}$, and this completes the proof. ∎

Acknowledgements

The work is partially supported by the National Natural Science Foundation of China under Grants 11371162, 11571134, and the Self-determined Research Funds of CCNU from the colleges basic research and operation of MOE.