The Upper Bound of the Edge Mostar Index with Respect to Bicyclic Graphs

Abstract

Let G be a connected graph; the edge Mostar index $M{o}_e\left(G\right)$ of G is defined as ${\displaystyle M{o}_e\left(G\right)=\sum _{e=uv\in E\left(G\right)}|m_u\left(e\right)-m_v\left(e\right)|,}$ where $m_u\left(e\right)$ and $m_v\left(e\right)$ denote the number of edges in G that are closer to vertex u than to vertex v and the number of edges that are closer to vertex v than to vertex u, respectively. In this paper, we determine the upper bound of the edge Mostar index for all bicyclic graphs and identify the extremal graphs that achieve this bound.

1. Introduction

The scope of the analysis carried out in the article concerns finite, undirected, and simple graphs. For terms and symbols, please refer to [1]. Let $C_m$ and $S_m$ denote cycles and stars with m edges, respectively. For a connected graph G, the set of vertices is denoted by $V\left(G\right)$ and the set of edges is denoted by $E\left(G\right)$. For a vertex $u\in V\left(G\right)$, the degree of u, denoted by $d_G\left(u\right)$ (or simply $d\left(u\right)$), is the number of edges which are incident to u. If $d\left(u\right)=1$, then the vertex u is called a pendant vertex of G. Furthermore, if either $d\left(u\right)=1$ or $d\left(v\right)=1$, then the edge $uv$ is called a pendant edge of G. For $u,v\in V\left(G\right),d(u,v)$ denote the distance between u and v in G, namely the length of the shortest path connecting vertex u and vertex v.

Chemical graph theory is one of the research directions of graph theory, and topological indices are the main research content of chemical graph theory. Topological indices are the invariants of the graph, which usually reflect the chemical and pharmacological properties of the molecule. Among all the topological indices, the most well known is the Wiener index. The definition is as follows:

$${\displaystyle W\left(G\right)=\sum _{\{u,v\}\subseteq V\left(G\right)}d(u,v).}$$

This topological index is widely studied in the field of chemical graph theory, see [2,3,4]. For $e=uv\in E\left(G\right)$, define the following set:

$$N_u\left(e\right)=\{w\in V\left(G\right):d(u,w)<d(v,w)\},$$

$$N_v\left(e\right)=\{w\in V\left(G\right):d(v,w)<d(u,w)\},$$

$$N_0\left(e\right)=\{w\in V\left(G\right):d(u,w)=d(v,w)\}.$$

Therefore, the sets $N_u\left(e\right), N_v\left(e\right)$, and $N_0\left(e\right)$ form a partition of the vertices in G. Let $n_u\left(e\right),n_v\left(e\right)$, and $n_0\left(e\right)$ denote the number of vertices in the three sets above, respectively. Evidently, if the graph G has n vertices, then $n_u\left(e\right)+n_v\left(e\right)+n_0\left(e\right)=n$.

A well-established property of the Wiener index is its formula [5]:

$${\displaystyle W\left(T\right)=\sum _{e=uv\in E\left(T\right)}{n}_u\left(e\right)n_v\left(e\right),}$$

which is applicable for a tree T. In order to generalize the above formula to the general graph G, Gutman [6] introduced a new topological index and subsequently called it the Szeged index:

$${\displaystyle Sz\left(G\right)=\sum _{e=uv\in E\left(G\right)}{n}_u\left(e\right)n_v\left(e\right).}$$

For any $e=uv\in E\left(G\right)$, we define the distance $d(e,x)$ from the edge e to the vertex x as:

$$d(e,x)=\min \left\{d\right(u,x),d(v,x\left)\right\}.$$

Similarly, based on the above definition, we define another three sets $M_u\left(e\right)$, $M_v\left(e\right)$, and $M_0\left(e\right)$ as follows:

$$M_u\left(e\right)=\{f\in E\left(G\right):d(u,f)<d(v,f)\},$$

$$M_v\left(e\right)=\{f\in E\left(G\right):d(v,f)<d(u,f)\},$$

$$M_0\left(e\right)=\{f\in E\left(G\right):d(u,f)=d(v,f)\}.$$

Let $m_u\left(e\right),m_v\left(e\right)$, and $m_0\left(e\right)$ denote the number of edges in the three sets above, respectively. Evidently, if the graph G has m edges, then $m_u\left(e\right)+m_v\left(e\right)+m_0\left(e\right)=m$.

Then, the edge Szeged index [7] of a graph G is defined by:

$${\displaystyle S{z}_e\left(G\right)=\sum _{e=uv\in E\left(G\right)}{m}_u\left(e\right)m_v\left(e\right),}$$

where $m_u\left(e\right)$ and $m_v\left(e\right)$ denote the number of edges in G that are closer to vertex u than to vertex v and the number of edges that are closer to vertex v than to vertex u, respectively.

Recently, Došlić et al. proposed a new topological index called the Mostar index, which was introduced in [8]. It can be written as:

$${\displaystyle Mo\left(G\right)=\sum _{e=uv\in E\left(G\right)}|n_u\left(e\right)-n_v\left(e\right)|.}$$

For results on the Mostar index, one can refer to [9,10,11,12,13].

Similarly, the edge Mostar index of a graph G is described as follows:

$${\displaystyle M{o}_e\left(G\right)=\sum _{e=uv\in E\left(G\right)}|m_u\left(e\right)-m_v\left(e\right)|.}$$

Example, for a path $P_3$ of length 3, denote its four vertices as $u_1$, $u_2$, $u_3$, $u_4$. For $e=u_1{u}_2$, we have $m_{u_1}\left(e\right)=0$, $m_{u_2}\left(e\right)=2$, then $|m_{u_1}\left(e\right)-m_{u_2}\left(e\right)|=2$. Then, for $e=u_2{u}_3$, we have $m_{u_2}\left(e\right)=1$, $m_{u_3}\left(e\right)=1$, then $|m_{u_2}\left(e\right)-m_{u_3}\left(e\right)|=0$. Finally, for $e=u_3{u}_4$, we have $m_{u_3}\left(e\right)=2$, $m_{u_4}\left(e\right)=0$, then $|m_{u_3}\left(e\right)-m_{u_4}\left(e\right)|=2$. So, according to the definition of the edge Mostar index, we obtain ${\displaystyle M{o}_e\left(P_3\right)=\sum _{e=uv\in E\left(P_3\right)}|m_u\left(e\right)-m_v\left(e\right)|=4.}$

With respect to the edge Mostar index, the extremal graphs among polymers [14], trees and unicyclic graphs [15], cacti graphs with fixed cycles [16], cycle-related graphs [17], and the minimum values of bicyclic graphs [18], have been studied.

In this paper, we provide an upper bound for the edge Mostar index of connected bicyclic graphs and characterize the graphs that achieve this upper bound.

Theorem 1.

Let G be a connected bicyclic graph with $m\ge 28$ edges. Then

$$M{o}_e\left(G\right)\le m^2-m-24,$$

the equality holds if and only if $G\cong B_0$ (see Figure 1).

2. Preparation

For the sake of convenience, let $\varphi \left(e\right)=|m_u\left(e\right)-m_v\left(e\right)|$; then, the definition of the edge Mostar index of a graph G can be further written as

$${\displaystyle M{o}_e\left(G\right)=\sum _{e=uv\in E\left(G\right)}|m_u\left(e\right)-m_v\left(e\right)|=\sum _{e\in E\left(G\right)}\varphi \left(e\right).}$$

A subgraph H of a graph G is isometric if the distance between any pair of vertices in H is the same as the distance between the same pair of vertices in G.

Lemma 1

([15]). Let $e=uv\in E\left(G\right)$. Then,

$$\varphi \left(e\right)=|m_u\left(e\right)-m_v\left(e\right)|\le m-1,$$

the equality holds if and only if $e=uv$ is a pendent edge.

Lemma 2.

Let $e=uv$ be a cut edge, but not a pendent edge, in the connected graph G. $G{}^{\prime }$ is obtained by contracting the edge e into a vertex w and adding a pendent edge $e{}^{\prime }=wz$ to this vertex, (see Figure 2). Then, $M{o}_e\left(G\right)<M{o}_e\left(G{}^{\prime }\right)$.

Proof. 

Let $|E\left(G_1\right)|=m_1$, $|E\left(G_2\right)|=m_2$ and $m_1\ge m_2\ge 1$. Then, based on the definition of the edge Mostar index, we know that ${\varphi }_G\left(e\right)=m_1-m_2$, ${\varphi }_{G{}^{\prime }}\left(e{}^{\prime }\right)=m_1+m_2$. The $\varphi \left(e\right)$ of other edges stays unchanged. Then,

$$M{o}_e\left(G\right)-M{o}_e\left(G{}^{\prime }\right)=m_1-m_2-(m_1+m_2)=-2m_2<0.$$

So, $M{o}_e\left(G\right)<M{o}_e\left(G{}^{\prime }\right)$. □

We now consider the graph $G\cong G_1·G_2$, where $G_1·G_2$ is the graph obtained from the $G_1$ and $G_2$ by identifying one vertex. Assume that w is common vertex of $G_1$ and $G_2$. Obviously, w is a cut vertex of the graph G. For each edge $e=uv\in E\left(G_1\right)$, the vertex w belongs to one of the three sets $N_u\left(e\right),N_v\left(e\right),N_0\left(e\right)$. Because every path connecting u(or v) and any vertex in $V\left(G_2\right)$ is must pass through w. So, all vertices of $G_2$ must be contained in one of the three sets $N_u\left(e\right),N_v\left(e\right),N_0\left(e\right)$. Then, we have that all the edges in $G_2$ belong to one of the three sets: $M_u\left(e\right),M_v\left(e\right),M_0\left(e\right)$. Thus, the contribution of $G_2$ to $\varphi \left(uv\right)=|m_u\left(e\right)-m_v\left(e\right)|$ depends entirely on the size of $G_2$, namely, changing the structure of $G_2$ does not change the value of ${\displaystyle \sum _{e=uv\in E\left(G_1\right)}|m_u\left(e\right)-m_v\left(e\right)|}$. Furthermore, because the pendent edge contributes the most to $\varphi \left(uv\right)=|m_u\left(e\right)-m_v\left(e\right)|$, we have the following lemma:

Lemma 3.

Let $H(\ncong S_m)$ be a connected graph with m edges. Then,

$$M{o}_e(G_1·S_m)>M{o}_e(G_1·H),$$

where the common vertex of $G_1·S_m$ is the center vertex of $S_m$.

Let $S_{m,r}\cong S_{m-r}·C_r$, where the common vertex w of $S_{m-r}·C_r$ is the center vertex of $S_{m-r}$, and we call w is the center vertex of $S_{m,r}$.

Lemma 4

([15]). Let G be an m-edge connected unicyclic graph. Then,

$$M{o}_e\left(G\right)\le \left\{\begin{array}{cc}{m}^2-2m-3,\hfill & 3\le m\le 8,\hfill \\ 60,\hfill & m=9,\hfill \\ m^2-m-12,\hfill & m\ge 10,\hfill \end{array}\right.$$

the equality holds if and only if $G\cong S_{m,3}$, for $3\le m\le 8$; $G\cong S_{m,3}$ or $S_{m,4}$, for $m=9$; $G\cong S_{m,4}$, for $m\ge 10$.

Lemma 5.

Let $H_1$ be a connected graph and $H_2$ be an unicyclic graph, where $|E\left(H_1\right)|=m_1$ and $|E\left(H_2\right)|=m_2$. Then, $M{o}_e(H_1·H_2)\le M{o}_e(H_1·S_{m_2,3})$, for $3\le m_1+m_2\le 8$; $M{o}_e(H_1·H_2)\le M{o}_e(H_1·S_{m_2,3})=M{o}_e(H_1·S_{m_2,4})$, for $m_1+m_2=9$; $M{o}_e(H_1·H_2)\le M{o}_e(H_1·S_{m_2,4})$, for $m_1+m_2\ge 10$, where the common vertex of $H_1·S_{m_2,3}$($H_1·S_{m_2,4}$) is the center vertex of $S_{m_2,3}$($S_{m_2,4}$).

Proof. 

If $3\le m_1+m_2\le 8$. Then, by Lemma 4, we get:

$$\begin{array}{ccc}\hfill M{o}_e(H_1·H_2)& =& {\displaystyle \sum _{e\in E(H_1·H_2)}\varphi \left(e\right)=\sum _{e\in E\left(H_1\right)}\varphi \left(e\right)+\sum _{e\in E\left(H_2\right)}\varphi \left(e\right)}\hfill \\ & =& {\displaystyle \sum _{e\in E\left(H_1\right)}\varphi \left(e\right)+M{o}_e(S_{m_1}·H_2)-m_1(m_1+m_2-1)}\hfill \\ & \le & {\displaystyle \sum _{e\in E\left(H_1\right)}\varphi \left(e\right)+M{o}_e(S_{m_1}·S_{m_2,3})-m_1(m_1+m_2-1)}\hfill \\ & =& {\displaystyle \sum _{e\in E\left(H_1\right)}\varphi \left(e\right)+\sum _{e\in E\left(S_{m_2,3}\right)}\varphi \left(e\right)}\hfill \\ & =& M{o}_e(H_1·S_{m_2,3}).\hfill \end{array}$$

Similarly, if $m_1+m_2=9$, then $M{o}_e(H_1·H_2)\le M{o}_e(H_1·S_{m_2,3})=M{o}_e(H_1·S_{m_2,4})$; and if $m_1+m_2\ge 10$, then $M{o}_e(H_1·H_2)\le M{o}_e(H_1·S_{m_2,4})$. □

3. Proof of Theorem 1

A theta graph $\theta (a,b,c)$ consists of two vertices, $v_1$ and $v_2$, which are connected by three internally disjoint paths $P_a,P_b$, and $P_c$, each with lengths $a,b$, and c, respectively. By symmetry, we can assume that $a\le b\le c$.

Let G be a bicyclic graph; then, $\left|E\right(G\left)\right|=\left|V\right(G\left)\right|+1$.

Based on the definition of bicyclic graphs and theta graphs, bicyclic graphs can be divided into two types [19]:

Let ${\mathcal{G}}_m^1$ be the set of connected bicyclic graphs with m-edge and exactly two cycles.

Let ${\mathcal{G}}_m^2$ be the set of connected bicyclic graphs with m-edge and three cycles; namely, if $G\in {\mathcal{G}}_m^2$, then it must have a subgraph isomorphic to $\theta (a,b,c)$.

Theorem 2.

Let $G\in {\mathcal{G}}_m^1$. If $m\ge 10$, then $M{o}_e\left(G\right)\le m^2-m-24$, and the equality holds if and only if $G\cong B_0$. If $m=9$, then $M{o}_e\left(G\right)\le 48$, and the equality holds if and only if $G\cong B_0,B_1,B_2$; If $6\le m\le 8$, then $M{o}_e\left(G\right)\le m^2-3m-6$, and the equality holds if and only if $G\cong B_2$ (see Figure 3).

Proof. 

Because $G\in {\mathcal{G}}_m^1$, then the graph G contains exactly two cycles. Furthermore, by Lemma 2, there are two unicyclic graphs $H_1$ and $H_2$ such that $G\cong H_1·H_2$. Suppose that $|E\left(H_1\right)|=m_1$, $|E\left(H_2\right)|=m_2$, then $\left|E\left(G\right)\right|=m=m_1+m_2$. By repeated using Lemma 5, we get

$$\begin{array}{c}\hfill M{o}_e\left(G\right)=M{o}_e(H_1·H_2)\le M{o}_e(H_1·S_{m_2,4})\le M{o}_e(S_{m_1,4}·S_{m_2,4})=M{o}_e\left(B_0\right),\end{array}$$

for $m_1+m_2\ge 10$;

$$\begin{array}{ccc}\hfill M{o}_e\left(G\right)& =& M{o}_e(H_1·H_2)\le M{o}_e(H_1·S_{m_2,3})=M{o}_e(H_1·S_{m_2,4})\hfill \\ & \le & M{o}_e(S_{m_1,3}·S_{m_2,3})=M{o}_e(S_{m_1,3}·S_{m_2,4})=M{o}_e(S_{m_1,4}·S_{m_2,4})\hfill \\ & =& M{o}_e\left(B_2\right)=M{o}_e\left(B_1\right)=M{o}_e\left(B_0\right),\hfill \end{array}$$

for $m_1+m_2=9$;

$$\begin{array}{c}\hfill M{o}_e\left(G\right)=M{o}_e(H_1·H_2)\le M{o}_e(H_1·S_{m_2,3})\le M{o}_e(S_{m_1,3}·S_{m_2,3})=M{o}_e\left(B_2\right),\end{array}$$

for $6\le m_1+m_2\le 8$.

Clearly, $M{o}_e\left(B_0\right)=m^2-m-24$ and $M{o}_e\left(B_2\right)=m^2-3m-6$. Therefore, the proof is complete. □

Now, we consider the bicyclic graphs referred to ${\mathcal{G}}_m^2$. For any edge $e=uv$ of a graph G, based on Lemma 1, we have

$$\varphi \left(e\right)=|m_u\left(e\right)-m_v\left(e\right)|\le m-1.$$

The equality holds if and only if $e=uv$ is a pendent edge. To determine the maximum value of the edge Mostar index, it is more convenient to consider the expression $(m-1)-\varphi \left(e\right)$, which we call the deficit of e. This deficit is denoted by $def\left(e\right)$.

For a graph G, the symbol $Def\left(G\right)$ represents the sum of the deficits of all edges in the graph G. By summing over all edges one obtains, $Def\left(G\right)+M{o}_e\left(G\right)=\left|E\left(G\right)\right|(m-1)$.

Lemma 6.

Let $m\ge 7$. Then, $M{o}_e\left(B_3\right)=m^2-m-28$ (see Figure 4).

Proof. 

In graph $B_3$, both non-pendent edges on the left have a deficit of 6, whereas the remaining non-pendent edges both have a deficit of 4. Therefore, the deficit of $B_3$ is 28. Thus, we obtain $M{o}_e\left(B_3\right)=m(m-1)-28=m^2-m-28$. □

Corollary 1.

Let $G\in {\mathcal{G}}_m^2$. If G is a graph with the maximum edge Mostar index, then $Def\left(G\right)\le 28$.

Lemma 7.

For any two edges $e=uv$ and f in a connected graph G, we have $\left|d\right(u,f)-d(v,f\left)\right|\le 1$.

Proof. 

Without loss of generality, assume that $d(u,f)=h$, then $d(v,f)\le h+1$. Furthermore, if $d(v,f)\le h-2$, then $d(u,f)=d(u,v)+d(v,f)\le h-1$, a contradiction. Thus, $d(v,f)\in [h-1,h+1]$. Hence, $\left|d\right(u,f)-d(v,f\left)\right|\le 1$. □

Lemma 8.

Let G be a connected bicyclic graph with m edges and an odd cycle. Then, we have $Def\left(G\right)\ge m+1$.

Proof. 

Let C be an odd cycle in G, and $f\in E\left(G\right)$ an arbitrary edge. we claim that there are two adjacent vertices $u_1$ and $u_2$ in C, such that $d(u_1,f)=d(u_2,f)$.

Suppose that this is not the case. Let a denote the minimum value of the distance between a vertex in the cycle C and the edge f, and let b denote the maximum value of the distance between a vertex in the cycle C and the edge f. Then, for any vertex in the cycle C whose distance from the edge f is k, we have $a\le k\le b$.

Let

$$X=\left\{x\right|x\in V\left(C\right)andd(x,f)=k,kiseven,a\le k\le b\},$$

$$Y=\left\{y\right|y\in V\left(C\right)andd(y,f)=k,kisodd,a\le k\le b\}.$$

By assumption, there are no two adjacent vertices in the cycle C with the same distance from the edge f; therefore, from Lemma 7, we can obtain $|d(u_1,f)-d(u_2,f)|=1$. So, every edge in C has one vertex in set X and the other vertex in set Y. This means that cycle C is a bipartite graph, which implies that it is an even-length cycle. This contradicts our assumption. Thus, there are two adjacent vertices $u_1$ and $u_2$ in C such that $d(u_1,f)=d(u_2,f)$.

Because $\varphi \left(u_1{u}_2\right)=|m_{u_1}\left(e\right)-m_{u_2}\left(e\right)|$ and $m_{u_1}\left(e\right)+m_{u_2}\left(e\right)+m_0\left(e\right)=m$, we obtain

$$\varphi \left(u_1{u}_2\right)=|m-m_0\left(e\right)-2m_{u_2}\left(e\right)|\le m-2.$$

This means that the deficit contribution of edge f to edge $u_1{u}_2$ is 1. Because edge f can be any edge of G, the sum of the deficits of edges in C is at least m. Because G is a bicyclic graph, it includes an edge e that is neither a pendent edge nor in C. According to Lemma 1, we have $\varphi \left(e\right)\le m-2$. Thus, the deficit of edge e is at least 1.

In conclusion, the deficit of a bicyclic graph G with an odd cycle is at least $m+1$. □

Theorem 3.

Let $G\in {\mathcal{G}}_m^2$. If $m\ge 28$, then $M{o}_e\left(G\right)\le m^2-m-28$, and the equality holds if and only if $G\cong B_3$.

Proof. 

Let G be a bicyclic graph containing $\theta (a,b,c)$ as a subgraph and with the maximum edge Mostar index. First, it follows from Lemma 2 that all edges in the graph G are pendent edges except for the edges contained in $\theta (a,b,c)$. Second, if graph G contains an odd cycle, then according to Lemma 8, $Def\left(G\right)\ge m+1\ge 29$, which contradicts Corollary 1. Therefore, graph G does not contain odd cycles. This implies that $a,b,c$ have the same parity.

By the definition of a theta graph, we have $a\le b\le c$. Let C be the shortest cycle in G. Now, take any edge $e=uv$ of C. Assume that path $P_c$ has $h_1$ edges closer to vertex u than to vertex v, and $h_2$ edges closer to vertex v than to vertex u, such that $h_1+h_2\le c$. In addition, let us assume that the graph G has p pendent edges that are closer to vertex u than to vertex v, and q pendent edges that are closer to vertex v than to vertex u. Furthermore, we have $p+q=m-a-b-c$. Because C is isometric, for $e=uv$, we have

$$\begin{array}{ccc}\hfill def\left(uv\right)& =& (m-1)-\varphi \left(uv\right)=(m-1)-|h_1+p-h_2-q|\hfill \\ & \ge & (a+b+c+p+q-1)-|h_1+p-h_2-q|\ge a+b-1.\hfill \end{array}$$

Furthermore, because the cycle C is even, $a+b$ is even and $a+b\ge 4$. If $a+b\ge 6$, then $Def\left(G\right)\ge 6\times 5=30>28$, which is a contradiction by Corollary 1. Thus, $a+b=4$. There are two cases.

Case 1: $a=1$ and $b=3$. Because G contains an isometric cycle of length $c+1$, we can conclude that $c=3$. Therefore, the graph G contains $\theta (1,3,3)$ as a subgraph. Let $v_1$ and $v_2$ be vertices of degree three in $\theta (1,3,3)$ of graph G, whereas $v_3,v_4,v_5,v_6$ are vertices of degree two. Because $\varphi \left(v_1{v}_2\right)\le m-7$, $\varphi \left(v_1{v}_3\right)\le m-4$, we have

$$def\left(v_1{v}_2\right)=(m-1)-\varphi \left(v_1{v}_2\right)\ge 6,def\left(v_1{v}_3\right)=(m-1)-\varphi \left(v_1{v}_3\right)\ge 3.$$

Similarly, $def\left(v_3{v}_4\right)\ge 6$, $def\left(v_5{v}_6\right)\ge 6$, $def\left(v_1{v}_5\right)\ge 3$, $def\left(v_2{v}_4\right)\ge 3$, $def\left(v_2{v}_6\right)\ge 3$. Then, for the graph G, $Def\left(G\right)\ge 3\times 6+4\times 3=30>28$, a contradiction with Corollary 1.

Case 2: $a=2$ and $b=2$. Because G contains an isometric cycle of length $c+2$, we can conclude that $c=2$. Therefore, the graph G contains $\theta (2,2,2)$ as a subgraph. Suppose that $v_1$ and $v_2$ are vertices of degree three in $\theta (2,2,2)$ of graph G, and $v_3,v_4,v_5$ are vertices of degree two. Let $l_i$ be the number of pendent edges of $v_i$ and $l_i\ge 0$.

If $l_1\ge 1$, we have $def\left(v_1{v}_3\right)\ge 4$, $def\left(v_1{v}_4\right)\ge 4$, $def\left(v_1{v}_5\right)\ge 4$, $def\left(v_2{v}_3\right)\ge 6$, $def\left(v_2{v}_4\right)\ge 6$, $def\left(v_2{v}_5\right)\ge 6$. Then, for the graph G, $Def\left(G\right)\ge 3\times 4+3\times 6=30>28$, a contradiction with Corollary 1. So, $l_1=0$. Similarly, there are also $l_2=0$.

If $l_3\ge 1$ and $l_4\ge 1$, we have $def\left(v_1{v}_3\right)\ge 6$, $def\left(v_1{v}_4\right)\ge 6$, $def\left(v_1{v}_5\right)\ge 4$, $def\left(v_2{v}_3\right)\ge 6$, $def\left(v_2{v}_4\right)\ge 6$, $def\left(v_2{v}_5\right)\ge 4$. Then, for the graph G, $Def\left(G\right)\ge 4\times 6+2\times 4=32>28$, a contradiction with Corollary 1. Therefore, all pendent edges in G must be associated with the same fixed vertex of degree two in $\theta (2,2,2)$.

Clearly, $G\cong B_3$ and $M{o}_e\left(B_3\right)=m^2-m-28$. Therefore, the proof is complete. □

Finally, Theorem 1 can be proved by Theorems 2 and 3.