On the Non-Commuting Graph of Dihedral Group

TL;DR

This paper studies various graph-theoretic properties of the non-commuting graph of dihedral groups, including indices, eccentricities, and mean distances, enhancing understanding of their structural characteristics.

Contribution

It introduces new calculations of indices and distances for the non-commuting graph of dihedral groups, expanding the algebraic graph theory literature.

Findings

Computed detour index, eccentric connectivity, and total eccentricity polynomials.

Determined the mean distance of the non-commuting graph for dihedral groups.

Provided explicit formulas for these graph invariants.

Abstract

For a nonabelian group G, the non-commuting graph $\Gamma_G$ of $G$ is defined as the graph with vertex set $G-Z(G)$, where $Z(G)$ is the center of $G$, and two distinct vertices of $\Gamma_G$ are adjacent if they do not commute in $G$. In this paper, we investigate the detour index, eccentric connectivity and total eccentricity polynomials of non-commuting graph on $D_{2n}$. We also find the mean distance of non-commuting graph on $D_{2n}$.

Equations

$$D_{2n}=⟨r,s:r^n=s^2=1,srs=r^{-1}⟩$$

$$|E(\Gamma_{\Omega})|=\left\{\begin{tabular}[]{ll}$\frac{3n(n-1)}{2}$,&\mbox{ if }$n$ is odd;\\ $\frac{3n(n-2)}{2}$,&\mbox{ if }$n$ is even.\\ \end{tabular}\right.$$

$$D(u,v)=\left\{\begin{tabular}[]{ll}$2n-2$,&\mbox{ if }$n$ is odd;\\ $2n-3$,&\mbox{ if }$n$ is even.\\ \end{tabular}\right.$$

$$D(\Gamma_{\Omega},x)=\left\{\begin{tabular}[]{ll}$(n-1)(2n-1)x^{2n-2}$,&\mbox{ if }$n$ is odd;\\ $(n-1)(2n-3)x^{2n-3}$,&\mbox{ if }$n$ is even.\\ \end{tabular}\right.$$

$$dd(\Gamma_{\Omega})=\left\{\begin{tabular}[]{ll}$2(n-1)^{2}(2n-1)$,&\mbox{ if }$n$ is odd;\\ $(n-1)(2n-3)^{2}$,&\mbox{ if }$n$ is even.\\ \end{tabular}\right.$$

$$ecc(v)=\left\{\begin{tabular}[]{ll}$2$,&\mbox{ if }$v\in\Omega_{1}$;\\ $1$,&\mbox{ if }$v\in\Omega_{2}$.\\ \end{tabular}\right.$$

$$\Xi(\Gamma_{\Omega},x)=\left\{\begin{tabular}[]{ll}$n(n-1)x^{2}+2n(n-1)x$,&\mbox{ if }$n$ is odd;\\ $3n(n-2)x^{2}$,&\mbox{ if }$n$ is even.\\ \end{tabular}\right.$$

$$\Theta(\Gamma_{\Omega},x)=\left\{\begin{tabular}[]{ll}$(n-1)x^{2}+nx$,&\mbox{ if }$n$ is odd;\\ $2(n-1)x^{2}$,&\mbox{ if }$n$ is even.\\ \end{tabular}\right.$$

$$\xi^{c}(\Gamma_{\Omega})=\left\{\begin{tabular}[]{ll}$4n(n-1)$,&\mbox{ if }$n$ is odd;\\ $6n(n-2)$,&\mbox{ if }$n$ is even.\\ \end{tabular}\right.$$

$$\sigma (r^i,{\Gamma }_{D_{2n}})={\Sigma }_{\begin{array}{c}1\le j<n\\ j\ne i\end{array}}d(r^i,r^j)+{\Sigma }_{1\le j\le n}d(r^i,s{r}^j)=2(n-2)+n=3n-4$$

$$\sigma (s{r}^i,{\Gamma }_{D_{2n}})={\Sigma }_{\begin{array}{c}1\le i,j\le n\\ i\ne j\end{array}}d(s{r}^i,s{r}^j)+{\Sigma }_{1\le j<n}d(s{r}^i,r^j)=(n-1)+(n-1)=2n-2,$$

$$\sigma (r^i,{\Gamma }_{D_{2n}})={\Sigma }_{\begin{array}{c}j\in S\\ j\ne i\end{array}}d(r^i,r^j)+{\Sigma }_{1\le j\le n}d(r^i,s{r}^j)=2(n-3)+n=3n-6$$

$$\sigma (s{r}^i,{\Gamma }_{D_{2n}})={\Sigma }_{\begin{array}{c}1\le j\le n\\ j\ne i\end{array}}d(s{r}^i,s{r}^j)+{\Sigma }_{j\in S}d(s{r}^i,r^j)=(n-2)+2+(n-2)=2n-2,$$

$$\sigma ({\Gamma }_{D_{2n}})$$

$$=(n-1)(3n-4)+n(2n-2)$$

$$=5n^2-9n+4.$$

$$\sigma ({\Gamma }_{D_{2n}})$$

$$=(n-2)(3n-6)+n(2n-2)$$

$$=5n^2-14n+12.$$

Full text

On the Non-Commuting Graph of Dihedral Group

S.M.S. Khasraw1, I.D. Ali2 and R.R. Haji3

1,2,3Department of Mathematics, College of Education,

Salahaddin University-Erbil, Erbil, Kurdistan Region, Iraq

1sanhan.khasraw@su.edu.krd, 2evan.ali1@su.edu.krd,

3rashad.haji@su.edu.krd

**Abstract

**

For a nonabelian group G, the non-commuting graph $\Gamma_{G}$ of $G$ is defined as the graph with vertex set $G-Z(G)$, where $Z(G)$ is the center of $G$, and two distinct vertices of $\Gamma_{G}$ are adjacent if they do not commute in $G$. In this paper, we investigate the detour index, eccentric connectivity and total eccentricity polynomials of non-commuting graph on $D_{2n}$. We also find the mean distance of non-commuting graph on $D_{2n}$.

1 Introduction

The concept of non-commuting graph of a finite group has been introduced by Abdollahi et al in 2006 [1]. For a non-abelian group $G$, associate a graph $\Gamma_{G}$ with it such that the vertex set of $\Gamma_{G}$ is $G-Z(G)$, where $Z(G)$ is the center of $G$, and two distinct vertices $x$ and $y$ are adjacent if they don’t commute in $G$, that is, $xy\neq yx$. Several works on assigning a graph to a group and investigation of algebraic properties of group using the associated graph have been done, for example, see [4, 5, 2].

All graphs are considered to be simple, which are undirected with no loops or multiple edges. Let $\Gamma$ be any graph, the sets of vertices and edges of $\Gamma$ are denoted by $V(\Gamma)$ and $E(\Gamma)$, respectively. The cardinality of the vertex set $V(\Gamma)$ is called the order of the graph $\Gamma$ and is denoted by $|V(\Gamma)|$ and the number of edges of the graph $\Gamma$ is called the size of $\Gamma$, and denoted by $|E(\Gamma)|$. The graph $\Gamma$ is called split if $V(\Gamma)=S\cup K$, where $S$ is an independent set and $K$ is a complete set. For a vertex $v$ in $\Gamma$, the number of edges incident to $v$ is called the degree of $v$ and is denoted by $deg_{\Gamma}(v)$. The eccentricity of a vertex $v$ in $\Gamma$, denoted by $ecc(v)$, is the largest distance between $v$ and any other vertex $u$ in $\Gamma$. For vertices $u$ and $v$ in a graph $\Gamma$, a $u-v$ path in $\Gamma$ is $u-v$ walk with no vertices repeated. The shortest (longest) $u-v$ path in a graph $\Gamma$, denoted by $d(u,v)$ $(D(u,v))$, is called the distance(detour distance) between vertices $u$ and $v$ in $\Gamma$. The detour index, *eccentric connectivity *and total eccentricity polynomials are defined as $D(\Gamma_{\Omega},x)=\sum_{u,v\in V(\Gamma)}x^{D(u,v)}$ [9], $\Xi(\Gamma,x)=\sum_{u\in V(\Gamma)}deg_{\Gamma}(u)x^{ecc(u)}$ and $\Theta(\Gamma,x)=\sum_{u\in V(\Gamma)}x^{ecc(u)}$ [7], respectively. The detour index $dd(\Gamma)$, the eccentric connectivity index and the total eccentricity $\xi^{c}(\Gamma)$ of a graph $\Gamma$ are the first derivatives of their corresponding polynomials at $x=1$, respectively. A transmission of a vertex $v$ in $\Gamma$ is $\sigma(v,\Gamma)=\Sigma_{u\in V(\Gamma)}d(u,v)$. The transmission of a graph $\Gamma$ is $\sigma(\Gamma)=\Sigma_{u\in V(\Gamma)}\sigma(u,\Gamma)$. The mean(average) distance of a graph $\Gamma$ is $\mu(\Gamma)=\frac{\sigma(\Gamma)}{p(p-1)}$, where $p$ is the order of $\Gamma$, see [8, 3, 6]. In this paper, we study some properties of non-commuting graph of dihedral groups. The dihedral group $D_{2n}$ of order $2n$ is defined by

[TABLE]

for any $n\geq 3$, and the center of $D_{2n}$ is Z(D_{2n})=\left\{\begin{tabular}[]{ll}{1},&\mbox{ if }n$is odd\\${1,r^{\frac{n}{2}}},&\mbox{ if }n is even.\\ \end{tabular}\right. Throughout this article, we assume that $\Omega_{1}=\{r^{i}:1\leq i\leq n\}-Z(D_{2n})$, and $\Omega_{2}=\{sr^{i}:1\leq i\leq n\}$. This article is organized as follows: In the present section, we give some important definitions and notations. In Section 2, we study some basic properties of the non-commuting graph $\Gamma_{D_{2n}}$ of $D_{2n}$. We see that $\Gamma_{D_{2n}}$ is a split graph if $n$ is an odd integer.

In Section 3, we find the detour index, eccentric connectivity and total eccentricity polynomials of the non-commuting graph $\Gamma_{D_{2n}}$. In Section 4, we find the mean distance of the graph $\Gamma_{D_{2n}}$.

2 Some properties of the non - commuting graph of $D_{2n}$

Recall that, for any $n\geq 3$, $D_{2n}=\langle r,s:r^{n}=s^{2}=1,srs=r^{-1}\rangle$, $\Omega_{1}=\{r^{i}:1\leq i\leq n\}-Z(D_{2n})$, and $\Omega_{2}=\{sr^{i}:1\leq i\leq n\}$.

We start with the following lemma, which has been proved in [1].

Lemma 2.1**.**

Let $G$ be any non-abelian finite group and $a$ be any vertex of $\Gamma_{G}$. Then $deg_{\Gamma_{G}}(a)=|G|-|C_{G}(a)|$, where $C_{G}(a)$ is the centralizer of the element $a$ in the group $G$.

According to the above lemma, we can state the following.

Theorem 2.2**.**

*In the graph $\Gamma_{\Omega}$, where $\Omega=\Omega_{1}\cup\Omega_{2}$, we have

$1.\;deg_{\Gamma_{\Omega}}(r^{i})=n$ for any $n$,

2.\;deg_{\Gamma_{\Omega}}(sr^{i})=\left\{\begin{tabular}[]{ll}2n-2,&\mbox{ if }n$is odd\\$2n-4,&\mbox{ if }n is even.\\ \end{tabular}\right.*

Proof.

1. Since $C_{D_{2n}}(r^{i})=\{r^{i}:1\leq i\leq n\}$, then, from Lemma 2.1, $deg_{\Gamma_{\Omega}}(r^{i})=|D_{2n}|-|C_{D_{2n}}(r^{i})|=2n-n=n$.

2. If $n$ is odd, then $C_{D_{2n}}(sr^{i})=\{1,sr^{i}\}$ for all $i$, $1\leq i\leq n$. This follows that $deg_{\Gamma_{\Omega}}(sr^{i})=2n-2$ for all $1\leq i\leq n$. If $n$ is even, then $C_{D_{2n}}(sr^{i})=\{1,r^{\frac{n}{2}},sr^{i},sr^{\frac{n}{2}+i}\}$ for all $1\leq i\leq n$. Thus, $deg_{\Gamma_{\Omega}}(sr^{i})=2n-4$ for all $1\leq i\leq n$. ∎

Theorem 2.3**.**

Let $\Gamma_{\Omega}$ be a non-commuting graph on $D_{2n}$.

If $\Omega=\Omega_{1}$, then $\Gamma_{\Omega}=\overline{K}_{l}$, where $l=|\Omega_{1}|$. 2. 2.

If $\Omega=\Omega_{2}$, then

\Gamma_{\Omega}=\left\{\begin{tabular}[]{ll}K_{n},&\mbox{ if }n$is odd\\$K_{n}-\frac{n}{2}K_{2},&\mbox{ if }n is even.\\ \end{tabular}\right.

where $\frac{n}{2}K_{2}$ denotes $\frac{n}{2}$ copies of $K_{2}$.

Proof.

1. The centralizer of $r^{i}$, $1\leq i\leq n$, is $C_{D_{2n}}(r^{i})=\{r^{i}:1\leq i\leq n\}$ of size $n$, then there is no edge between any pair of vertices in $\Gamma_{\Omega_{1}}$. Thus, $\Gamma_{\Omega_{1}}=\overline{K}_{l}$, where $l=|\Omega_{1}|$.

2. When $n$ is odd. Since the element $sr^{i}$, where $i=1,2,...,n$, has centralizer $C_{D_{2n}}(sr^{i})=\{1,sr^{i}\}$ of size 2, so let $\Omega=\Omega_{2}=\{sr,sr^{2},...,sr^{n}\}$. Then the subgraph $\Gamma_{\Omega}=K_{n}$ is complete.

When $n$ is even. Since $C_{D_{2n}}(sr^{i})=\{1,r^{\frac{n}{2}},sr^{i},sr^{\frac{n}{2}+i}\}$ for all $1\leq i\leq n$. Then there is no edge between the vertices $sr^{i}$ and $sr^{\frac{n}{2}+i}$ in $\Gamma_{\Omega}$ for all $1\leq i\leq n$. Therefore, $\Gamma_{\Omega}=K_{n}-\frac{n}{2}K_{2}$ ∎

Theorem 2.4**.**

Let $n\geq 3$ be an odd integer and $H$ be a subset of $D_{2n}-Z(D_{2n})$. Then $\Gamma_{H}=K_{1,n-1}$ if and only if $H=\{sr^{i},r,r^{2},\cdots,r^{n-1}\}$ for some $i$.

Proof.

Suppose that $\Gamma_{H}=K_{1,n}$. By Theorem 2.2, $H=\{sr^{i},r,r^{2},\cdots,r^{n-1}\}$ for some $i$. Conversely, suppose $H=\{sr^{i},r,r^{2},\cdots,r^{n-1}\}$. Then $C_{H}(sr^{i})=\{sr^{i}\}$ and $C_{H}(r^{j})=\{r,r^{2},\cdots,r^{n-1}\}$ for $1\leq j\textless n$. Thus, $\Gamma_{H}=K_{1,n-1}$. ∎

Corollary 2.5**.**

Let $n\geq 3$ be an odd integer and $\Omega=\Omega_{1}\cup\Omega_{2}$. Then $\Gamma_{\Omega}$ is a split graph.

Proof.

The proof follows from Theorem 2.3 and Theorem 2.4. ∎

Theorem 2.6**.**

Let $\Gamma_{\Omega}$ be a non-commuting graph on $D_{2n}$, where $\Omega=\Omega_{1}\cup\Omega_{2}$. We have

[TABLE]

Proof.

It is clear that $\Omega_{1}\cap\Omega_{2}=\emptyset$ and $\Omega_{1}\cup\Omega_{2}=D_{2n}-Z(D_{2n})=\Omega$. According to $n$, there are two cases to consider.

Case 1. If $n$ is odd, then the subgraph induced by $\Omega_{1}$ has no edges and the subgraph induced by $\Omega_{2}$ is complete. Thus, the number of edges in $\Gamma_{\Omega}$ is sum of the number of edges in $\langle\Omega_{2}\rangle$ and the number of edges from set of vertices in $\Omega_{1}$ to set of vertices in $\Omega_{2}$. Therefore, $|E(\Gamma_{\Omega})|=\frac{n(n-1)}{2}+n(n-1)=\frac{3n(n-1)}{2}$.

Case 2. If $n$ is even, then the subgraph induced by $\Omega_{1}$ has no edges and the subgraph induced by $\Omega_{2}$ has $\frac{n(n-1)}{2}-\frac{n}{2}=\frac{n(n-2)}{2}$ edges. Thus, the number of edges in $\Gamma_{\Omega}$ is sum of the number of edges in $\langle\Omega_{2}\rangle$ and the number of edges from set of vertices in $\Omega_{1}$ to set of vertices in $\Omega_{2}$. Therefore, $|E(\Gamma_{\Omega})|=\frac{n(n-2)}{2}+n(n-2)=\frac{3n(n-2)}{2}$. ∎

3 Detour index, eccentric connectivity and total eccentricity polynomials of non- commuting graphs on $D_{2n}$

Theorem 3.1**.**

Let $\Gamma_{\Omega}$ be a non-commuting graph on $D_{2n}$, where $\Omega=\Omega_{1}\cup\Omega_{2}$. Then for any $u,v\in\Gamma_{\Omega}$,

[TABLE]

Proof.

There are two cases. When $n$ is odd. From Theorem 2.3 and Theorem 2.4, we see that no two vertices in $\Omega_{1}$ are adjacent, any pair of distinct vertices in $\Omega_{2}$ are adjacent, and each vertex in $\Omega_{1}$ is adjacent to every vertex in $\Omega_{2}$. Then for all $u,v\in\Omega$, there is a $u-v$ path of length $2n-2$.

When $n$ is even. Again, no two vertices in $\Omega_{1}$ are adjacent, each vertex in $\Omega_{1}$ is adjacent to every vertex in $\Omega_{2}$, and any pair of distinct vertices $u$ and $v$ in $\Omega_{2}$ are adjacent if $u,v\notin\{sr^{i},sr^{\frac{n}{2}+i}\}$ for $1\leq i\leq\frac{n}{2}$. So, for all $u,v\in\Omega$, there is a $u-v$ path of length $2n-3$. ∎

Theorem 3.2**.**

Let $\Gamma_{\Omega}$ be a non-commuting graph on $D_{2n}$, where $\Omega=\Omega_{1}\cup\Omega_{2}$. Then

[TABLE]

Proof.

Case 1. n is odd. Since $|\Gamma_{\Omega}|=2n-1$, there are ${{2n-1}\choose{2}}=(n-1)(2n-1)$ possibilities of distinct pairs of vertices. By Theorem 3.1, $D(u,v)=2n-2$ for any $u,v\in\Gamma_{\Omega}$. Then $D(\Gamma_{\Omega},x)=\sum_{\{u,v\}}x^{D(u,v)}={{2n-1}\choose{2}}x^{2n-2}=(n-1)(2n-1)x^{2n-2}$.

Case 2. n is even. We have that $|\Gamma_{\Omega}|=2n-2$ and the possibility of taking distinct pairs of vertices form $\Gamma_{\Omega}$ is ${{2n-2}\choose{2}}=(n-1)(2n-3)$. From Theorem 3.1, we deduce that $D(\Gamma_{\Omega},x)=\sum_{\{u,v\}}x^{D(u,v)}={{2n-2}\choose{2}}x^{2n-3}=(n-1)(2n-3)x^{2n-3}$. ∎

Corollary 3.3**.**

For the graph $\Gamma_{\Omega}$,

[TABLE]

Proof.

It is clear that $dd(\Gamma_{\Omega})=\frac{d}{dx}(D(\Gamma_{\Omega},x))|_{x=1}$. From Theorem 3.2, the result follows. ∎

Theorem 3.4**.**

Let $\Gamma_{\Omega}$ be a non-commuting graph on $D_{2n}$, where $\Omega=\Omega_{1}\cup\Omega_{2}$.

When $n$ is odd, then

[TABLE] 2. 2.

When $n$ is even, then $ecc(v)=2$ for each $v\in\Omega$.

Proof.

1. When $n$ is odd. There is no edge between any pair of vertices in $\Omega_{1}$ and each vertex in $\Omega_{2}$ is adjacent to every vertex in $\Omega$. So the maximum distance between any vertex of $\Omega_{1}$ and the other vertices in $\Omega$ is 2 and the maximum distance between any vertex of $\Omega_{2}$ and the other vertices in $\Omega$ is 1.

2. When $n$ is even. Again, There is no edge between any pair of vertices in $\Omega_{1}$. Also, each vertex in $\Omega_{1}$ is adjacent to every vertex in $\Omega_{2}$. Thus, $ecc(v)=2$ for each $v\in\Omega_{1}$. By Theorem 2.3, the subgraph $\Gamma_{\Omega_{2}}$ is not a complete graph because there is no edge between the vertices $sr^{i}$ and $sr^{i+\frac{n}{2}}$. This means that the maximum distance between any vertex in $\Omega_{2}$ and any other vertex in $\Omega$ is 2, so $ecc(v)=2$ for each $v\in\Omega_{2}$. ∎

From above theorem, we can have the following.

Theorem 3.5**.**

Let $\Gamma_{\Omega}$ be a non-commuting graph on $D_{2n}$, where $\Omega=\Omega_{1}\cup\Omega_{2}$. Then

[TABLE] 2. 2.

[TABLE]

Proof.

The proof follows directly from Theorem 2.2 and Theorem 3.4. ∎

From the above theorem, one can obtain the eccentric connectivity index and the total eccentricity of a graph $\Gamma_{\Omega}$ from their corresponding polynomials by computing their first derivatives at $x=1$.

Corollary 3.6**.**

Let $\Gamma_{\Omega}$ be a non-commuting graph on $D_{2n}$, where $\Omega=\Omega_{1}\cup\Omega_{2}$. Then

[TABLE]

4 The mean distance of the graph $\Gamma_{D_{2n}}$

Through this section we find the mean (average) distance of the graph $\Gamma_{D_{2n}}$.

Lemma 4.1**.**

In the graph $\Gamma_{D_{2n}}$, where $n$ is odd, the transmission of each vertex $r^{i}$ is $\sigma(r^{i},\Gamma_{D_{2n}})=3n-4$ for all $1\leq i\leq n-1$ and the transmission of a vertex $sr^{i}$ is $\sigma(sr^{i},\Gamma_{D_{2n}})=2n-2$ for all $1\leq i\leq n$.

Proof.

The vertices set of the graph $\Gamma_{D_{2n}}$ is $V(\Gamma_{D_{2n}})=\{r^{i},sr^{j}:1\leq i<n,1\leq j\leq n\}$. Then $|V(\Gamma_{D_{2n}})|=2n-1$, where $n$ is odd. A vertex $r^{i}$ is adjacent with all vertices $sr^{j}$ for all $1\leq j\leq n$, so, $d(r^{i},sr^{j})=1$ for all $1\leq i\leq n-1$ and all $1\leq j\leq n$. While a vertex $r^{i}$ is not adjacent with $r^{j}$ for all $i\neq j$, $1\leq i\leq n-1$ and $1\leq j\leq n$, then $d(r^{i},r^{j})=2$ for all $1\leq i\leq n-1$, $1\leq j\leq n$ and $i\neq j$. So,

[TABLE]

for all $1\leq i\leq n-1$. On the other hand every vertex $sr^{i}$ is adjacent with $sr^{j}$ for all $i\neq j$, $1\leq i,j\leq n$. Therefore, $d(sr^{i},sr^{j})=1$, for all $i\neq j$, $1\leq i,j\leq n$. Also, every vertex $sr^{i}$ is adjacent with $r^{j}$, then $d(sr^{i},r^{j})=1$ for all $1\leq i\leq n$, $1\leq j\leq n-1$. So,

[TABLE]

for all $1\leq i\leq n$. ∎

Lemma 4.2**.**

In the graph $\Gamma_{D_{2n}}$, where $n$ is even, the transmission of each vertex $r^{i}$ is $\sigma(r^{i},\Gamma_{D_{2n}})=3n-6$ for all $1\leq i\leq n-1$ and the transmission of a vertex $sr^{i}$ is $\sigma(sr^{i},\Gamma_{D_{2n}})=2n-2$ for all $1\leq i\leq n.$

Proof.

Let $M=\{1,2,\ldots,n-1\}-\{n/2\}$. Then the vertices set of the graph $\Gamma_{D_{2n}}$, where $n$ is even, is $V(\Gamma_{D_{2n}})=\{r^{i},sr^{j}:i\in M,1\leq j\leq n\}$. So, $|V(\Gamma_{D_{2n}})|=2n-2$. A vertex $r^{i}$ is adjacent with all vertices $sr^{j}$ for all $i\in M$ and all $1\leq j\leq n$. Thus, $d(r^{i},sr^{j})=1$ for all $i\in M$ and all $1\leq j\leq n$. Notice that every two vertices $r^{i}$ and $r^{j}$ are non-adjacent for all $i,j\in M$ and $i\neq j$, then $d(r^{i},r^{j})=2$ for all $i,j\in M$ and $i\neq j$. So,

[TABLE]

for all $i\in M$. Also, every vertex $sr^{i}$ is adjacent with $sr^{j}$ for all $i\neq j$, $1\leq i\leq n/2$, and all $j\in\{1,2,\ldots,n-1\}-\{i+n/2\}$, then $d(sr^{i},sr^{j})=1$, for all $j\in\{1,2,\ldots,n-1\}-\{i+n/2\}$, and $d(sr^{i},sr^{i+n/2})=2$, for all $1\leq i\leq n/2$. Since each vertex $sr^{i}$ is adjacent with all vertices $r^{j}$, for all $1\leq i\leq n$, and $j\in M$, then $d(sr^{i},r^{j})=1$. Therefore,

[TABLE]

for all $1\leq i\leq n$. ∎

Theorem 4.3**.**

The mean distance of the graph $\Gamma_{D_{2n}}$, where $n$ is odd, is $\mu(\Gamma_{D_{2n}})=\frac{5n-4}{4n-2}$.

Proof.

By Lemma 4.1, we see that the transmission of the graph $\Gamma_{D_{2n}}$ is

[TABLE]

Notice that $|V(\Gamma_{D_{2n}})|=2n-1$. Therefore, $\mu(\Gamma_{D_{2n}})=\frac{\sigma(\Gamma_{D_{2n}})}{|V(\Gamma_{D_{2n}})|(|V(\Gamma_{D_{2n}})|-1)}=\frac{5n^{2}-9n+4}{(2n-1)(2n-2)}=\frac{5n-4}{4n-2}$. ∎

Theorem 4.4**.**

The mean distance of the graph $\Gamma_{D_{2n}}$, where $n$ is even, is $\mu(\Gamma_{D_{2n}})=\frac{5n^{2}-14n+12}{(2n-2)(2n-3)}$.

Proof.

By using Lemma 4.2, we can find the transmission of the graph $\Gamma_{D_{2n}}$ which is

[TABLE]

Notice that $|V(\Gamma_{D_{2n}})|=2n-2$. Therefore, $\mu(\Gamma_{D_{2n}})=\frac{\sigma(\Gamma_{D_{2n}})}{|V(\Gamma_{D_{2n}})|(|V(\Gamma_{D_{2n}})|-1)}=\frac{5n^{2}-14n+12}{(2n-2)(2n-3)}$. ∎