The maximum Mostar indices of unicyclic graphs with given diameter

Highlights

• The greatest Mostar index among the graphs with order $n$ and diameter $d$ is determined.

• The graphs with the greatest Mostar index among the graph set above are determined.

• Cut method and graph transformations are used.

Abstract

For an edge $e=uv$ in a given graph $G$, let $n_G^u\left(e\right)$ be the number of vertices which have a less distance from $u$ than that from $v$. Then $|n_G^u\left(e\right)-n_G^v\left(e\right)|$ is called the contribution of $e$. The Mostar index is defined as the sum of the edge contributions in $G$. In this paper, the unicyclic graphs with order $n$ and diameter $d$, having the greatest Mostar index are determined.

Introduction

Let $G=(V_G,E_G)$ be a simple and undirected graph. Define $\left|G\right|=|V_G|$ being the order of $G$. Denote by $dis{t}_G(x,y)$ (or $dist(x,y)$ for short) the distance between two vertices $x,y$ in $G$, which is defined as the edge number of the shortest path between them. For each edge $e=uv\in E_G$, let

$$N_G^u\left(e\right)=\{x\in V_G|dist(x,u)<dist(x,v)\},$$

and $n_G^u\left(e\right)=\left|N_G^u\left(e\right)\right|$. Then

$${\varphi }_G\left(e\right)=|n_G^u\left(e\right)-n_G^v\left(e\right)|$$

is called the contribution of $e$. The Mostar index $Mo\left(G\right)$ of $G$ is defined as the sum of all edge contributions. That is,

$$Mo\left(G\right)=\sum _{e\in E_G}{\varphi }_G\left(e\right).$$

The Mostar index was proposed in [1], which measures peripherality in chemistry and complex networks, and measures how far a graph is from distance-balanced [2], [3], [4], [5], [6], where a distance-balanced graph is the graph with the Mostar index being zero. With respect to the index, the extremal graphs among trees [1], [7], [8], tree-like chemical graphs [9], [10], [11], [12], [13], [14], unicyclic graphs [1], bicyclic graphs [15], cacti [16] and so on, were studied. In [8], the Mostar index was found to be a good predictor for the total surface area of octane isomers, by being compared with other distance-based graph invariants. A recent article [17] gathered many known extremal results concerning the index, and stated some modifications of the index, such as the edge Mostar index [18], [19], [20], [21].

A unicyclic graph is the graph with exactly one cycle. The diameter of a graph is the greatest distance between two of its vertices. Let ${\mathcal{G}}_{n,d}$ be the graph set with order $n$ and diameter $d.$ For a graph set $\mathcal{G}$, let ${\mathcal{G}}^{\max }\subseteq \mathcal{G}$ be the graph set with the greatest Mostar index. The aim of the paper is to determine ${\mathcal{G}}_{n,d}^{\max }$ and their Mostar indices.

A cycle (resp. a path) with $k$ edges is called a $k$-cycle (resp. a $k$-path) for short. Let $G_{n,d,1,3}$, $G_{n,d,1,4}$, $G_{n,d,2,3}$, $G_{n,d,2,4}$ and $G_{n,d,3}$ be the graphs obtained by the following steps, respectively.

• $G_{n,d,1,3}$ (resp. $G_{n,d,1,4}$): Let $C$ be a 3-cycle (resp. a 4-cycle) and $v$ be a vertex in $C$. First, attach $(n-d-3)$ (resp. $(n-d-4)$) pendent edges at $v$. Second, attach a pendent $(\lceil d/2\rceil )$-path and a pendent $(\lfloor d/2\rfloor )$-path at $v$. See Fig. 1.

• $G_{n,d,2,3}$: Let $C$ be a 3-cycle and $uv$ be an edge in $C$. First, attach $(n-d-2)$ pendent edges at $v$. Second, if $d$ is odd, then either attach a pendent $\left[\right(d+1)/2]$-path at $v$ and a pendent $\left[\right(d-3)/2]$-path at $u$, or attach a pendent $\left[\right(d-1)/2]$-path at each of $u$ and $v$; see Fig. 2(a). If $d$ is even, then attach a pendent $(d/2)$-path at $v$ and a pendent $(d/2-1)$-path at $u$; see Fig. 2(b).

• $G_{n,d,2,4}$: Let $C$ be a 4-cycle and $uv$ be an edge in $C$. First, attach $(n-d-3)$ pendent edges at $v$. Second, if $d$ is odd, then attach a pendent $\left[\right(d+1)/2]$-path at $v$ and a pendent $\left[\right(d-3)/2)]$-path at $u$; see Fig. 3(a). If $d$ is even, then either attach a pendent $(d/2)$-path at $v$ and a pendent $(d/2-1)$-path at $u$, or attach a pendent $(d/2+1)$-path at $v$ and a pendent $(d/2-2)$-path at $u$; see Fig. 3(b).

• $G_{n,d,3}$: Let $C$ be a 4-cycle and $uv$ be an edge in $C$. First, attach a pendent $(d-2)$-path $P$ at $v$. Second, if $d=3,4,5$, then attach $(n-d-2)$ pendent edges at $v$; see Fig. 4(a). If $d\ge 6$ and $d$ is even, then attach $(n-d-2)$ pendent edges at $w$ in $P$ where $dis{t}_P(w,v)=d/2-2$ or $d/2-3$; see Fig. 4(b). If $d\ge 7$ and $d$ is odd, then attach $(n-d-2)$ pendent edges at $w$ in $P$ where $dis{t}_P(w,v)=(d-5)/2$; see Fig. 4(c).

Theorem 1.1

Suppose $n\ge 5$ and $3\le d\le n-2$. Let $G$ be a unicyclic graph with order $n$ and diameter $d$.

• If $d\ge 6$ and $n\ge d+4$, then

  $$Mo\left(G\right)\le n^2-2n-\lceil \frac{d^2}{2}\rceil +d-8,$$

  with the equality if and only if $G\cong G_{n,d,1,4}$ when $d\ge 8$; and $G\cong G_{n,d,1,4}$ or $G_{n,d,3}$ when $d\in \{6,7\}$.

• If $d\ge 6$ and $n\in \{d+2,d+3\}$, or $d\in \{4,5\}$, then

  $$Mo\left(G\right)\le n^2-2n-\lfloor \frac{d^2}{2}\rfloor -2,$$

  with the equality only if $G\cong G_{n,d,3}$ or $G_{n,d,2,3}$ when $d$ is even and $n=d+2$; and $G\cong G_{n,d,3}$ otherwise.

• Let $d=3.$ If $n=5$, then $Mo\left(G\right)\le 8$ with the equality if and only if $G\cong G_{5,3,2,3}$. If $n\ge 6$, then

  $$Mo\left(G\right)\le n^2-2n-8,$$

  with the equality if and only if $G\cong G_{n,3,1,3},G_{n,3,2,3}$ or $G_{n,3,3}$ when $n=6$; and $G\cong G_{n,3,3}$ when $n\ge 7$.

Theorem 1.1 is proved in Section 3. The proof idea is as follows: first, show ${\mathcal{G}}_{n,d}^{\max }\subseteq {\mathcal{G}}_0=\{G_{n,d,1,3},G_{n,d,1,4},G_{n,d,2,3},G_{n,d,2,3},G_{n,d,3}\}$; second, determine ${\mathcal{G}}_0^{\max }$ by straightforward calculations. The calculations are based on basic cut method, which is stated in Section 2. General cut method was already used in counting Mostar index [22] recently and in counting the closely related Szeged index [23] before. For more about cut method, one can also be referred to the latest survey paper [24]. The other method we use in the proof is doing some graph transformations, such as the moving operation which is also stated in Section 2.