Sanhan Muhammad Salih Khasraw^∗ , Ivan Dler Ali, Rashad Rashid Haji

Department of Mathematics, College of Education, Salahaddin University-Erbil, Erbil, Kurdistan Region, Iraq

sanhan.khasraw@su.edu.krd, evan.ali1@su.edu.krd, rashad.haji@su.edu.krd

For a nonabelian group G, the non-commuting graph Γ^G of G is defined as the graph with vertexset G−Z(G), where Z(G) is the center of G, and two distinct vertices of Γ^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 the non-commuting graph on D2n. We also find the mean distance of the non-commuting graph on D2n.

Keywords: dihedral group, non-commuting graph, detour distance, mean distance

Mathematics Subject Classification : 05C12, 05C25, 05C31, 05C40

DOI: 10.5614/ejgta.2020.8.2.3

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 Γ^G with it such that the vertex-set of Γ^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 6= 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 [3, 7, 8, 12, 6, 2].

All graphs are considered to be simple, which are undirected with no loops or multiple edges. Let Γ be any graph, the sets of vertices and edges of Γ are denoted by V (Γ) and E(Γ), respectively. The cardinality of the vertex-set V (Γ) is called the order of the graph Γ and is denoted by |V (Γ)| and the number of edges of the graph Γ is called the size of Γ, and denoted by |E(Γ)|. The graph Γ is called split if V (Γ) = S ∪ K, where S is an independent set and the subgraph induced

Received: 20 February 2019, Revised: 1 March 2020, Accepted: 29 March 2020.

^∗Corresponding author

by K is a complete graph. 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 \(D(\Gamma_{\Omega},x) = \sum_{u,v \in V(\Gamma)} x^{D(u,v)}\) [11], \(\Xi(\Gamma,x) = \sum_{u \in V(\Gamma)} deg_{\Gamma}(u)x^{ecc(u)}\) and \(\Theta(\Gamma,x) = \sum_{u \in V(\Gamma)} x^{ecc(u)}\) [10], 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 \(\sigma(v,\Gamma) = \sum_{u \in V(\Gamma)} d(u,v)\). The transmission of a graph \(\Gamma\) is \(\sigma(\Gamma) = \sum_{u \in V(\Gamma)} \sigma(u,\Gamma)\). The mean (average) distance of a graph \(\Gamma\) is \(\sigma(\Gamma) = \frac{\sigma(\Gamma)}{p(p-1)}\), where p is the order of \(\Gamma\), see [4, 5, 9]. 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

\[D_{2n} = \langle r, s : r^n = s^2 = 1, srs = r^{-1} \rangle\]

for any \(n \geq 3\), and the center of \(D_{2n}\) is \(Z(D_{2n}) = \begin{cases} \{e\}, & \text{if } n \text{ is odd;} \\ \{e, r^{\frac{n}{2}}\}, & \text{if } n \text{ is even.} \end{cases}\) 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 \(\Gamma_{\Omega}\) of \(D_{2n}\). We see that \(\Gamma_{\Omega}\) 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 \(\Gamma_{\Omega}\). In Section 4, we find the mean distance of the graph \(\Gamma_{\Omega}\).

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 2.1. In the graph \(\Gamma_{\Omega}\), where \(\Omega = \Omega_1 \cup \Omega_2\), we have

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

\[2. \deg_{\Gamma_{\Omega}}(r) = n \text{ for any } n,\] \[2. \deg_{\Gamma_{\Omega}}(sr^{i}) = \begin{cases} 2n - 2, & \text{if } n \text{ is odd;} \\ 2n - 4, & \text{if } n \text{ is even.} \end{cases}\]

Proof. 1. Since \(C_{D_{2n}}(r^i) = \{r^i : 1 \le i \le n\}\), 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 \(C_{D_{2n}}(sr^i) = \{e, sr^i\}\) for all \(i, 1 \le i \le n\). This follows that \(deg_{\Gamma_0}(sr^i) = 2n-2\)for all \(1 \leq i \leq n\). If n is even, then \(C_{D_{2n}}(sr^i) = \{e, r^{\frac{n}{2}}, sr^i, sr^{\frac{n}{2}+i}\}\) for all \(1 \leq i \leq n\). Thus, \(deg_{\Gamma_{\Omega}}(sr^{i}) = 2n - 4\) for all \(1 \leq i \leq n\).

Theorem 2.2. Let \(\Gamma_{\Omega}\) be a non-commuting graph on \(D_{2n}\).

• 1. If \(\Omega = \Omega_1\), then \(\Gamma_{\Omega} = \overline{K}_l\), where \(l = |\Omega_1|\).

1. If \[\Omega = \Omega_1\], then \(\Omega = K_1\), where \(t = 0\)
2. If \(\Omega = \Omega_2\), then \[\Gamma_{\Omega} = \begin{cases} K_n, & \text{if } n \text{ is odd;} \\ K_n - \frac{n}{2}K_2, & \text{if } n \text{ is even.} \end{cases}\]

where \(\frac{n}{2}K_2\) denotes \(\frac{n}{2}\) copies of \(K_2\).

Proof. 1. The centralizer of \(r^i\), \(1 \le i \le n\), is \(C_{D_{2n}}(r^i) = \{r^i : 1 \le i \le n\}\) 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 \(C_{D_{2n}}(sr^i)=\)\(\{e, sr^i\}\) 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 \(C_{D_{2n}}(sr^i) = \{e, r^{\frac{n}{2}}, sr^i, sr^{\frac{n}{2}+i}\}\) for all \(1 \le i \le n\). 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 2.3. Let \(n \ge 3\) 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 2.1, \(H=\{sr^i,r,r^2,\cdots,r^{n-1}\}\) for some i. Conversely, suppose \(H = \{sr^i, r, r^2, \cdots, r^{n-1}\}\). Then \(C_H(sr^i) = \{sr^i\}\) and \(C_H(r^j) = \{r, r^2, \cdots, r^{n-1}\}\). \(r^{n-1}\) for \(1 \le j < n\). Thus, \(\Gamma_H = K_{1,n-1}\).

Corollary 2.1. 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 2.2 and Theorem 2.3.

Theorem 2.4. Let \(\Gamma_{\Omega}\) be a non-commuting graph on \(D_{2n}\), where \(\Omega = \Omega_1 \cup \Omega_2\). We have

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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 \(\mathcal{D}_{2n}\)

Theorem 3.1. Let \(\Gamma_{\Omega}\) be a non-commuting graph on \(D_{2n}\), where \(\Omega = \Omega_1 \cup \Omega_2\). Then for any \(u, v \in \Gamma_{\Omega}\),

\[D(u,v) = \begin{cases} 2n-2, & \text{if } n \text{ is odd;} \\ 2n-3, & \text{if } n \text{ is even.} \end{cases}\]

Proof. There are two cases. When n is odd. From Theorem 2.2 and Theorem 2.3, 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 \(1 \le i \le \frac{n}{2}\). 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

\[D(\Gamma_{\Omega}, x) = \begin{cases} (n-1)(2n-1)x^{2n-2}, & \text{if } n \text{ is odd;} \\ (n-1)(2n-3)x^{2n-3}, & \text{if } n \text{ is even.} \end{cases}\]

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

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 \(\binom{2n-2}{2} = (n-1)(2n-3)\). From Theorem 3.1, we deduce that \(D(\Gamma_{\Omega}, x) = \sum_{\{u,v\}} x^{D(u,v)} = \binom{2n-2}{2} x^{2n-3} = (n-1)(2n-3)x^{2n-3}\).

Corollary 3.1. For the graph \(\Gamma_{\Omega}\),

\[dd(\Gamma_{\Omega}) = \begin{cases} 2(n-1)^2(2n-1), & \text{if } n \text{ is odd;} \\ (n-1)(2n-3)^2, & \text{if } n \text{ is even.} \end{cases}\]

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

Theorem 3.3. Let \(\Gamma_{\Omega}\) be a non-commuting graph on \(D_{2n}\), where \(\Omega = \Omega_1 \cup \Omega_2\).

1. When n is odd, then

\[ecc(v) = \begin{cases} 2, & \text{if } v \in \Omega_1; \\ 1, & \text{if } v \in \Omega_2. \end{cases}\]

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 2.2, 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 the above theorem, we can have the following.

Theorem 3.4. Let \(\Gamma_{\Omega}\) be a non-commuting graph on \(D_{2n}\), where \(\Omega = \Omega_1 \cup \Omega_2\). Then

1.

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2.

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Proof. The proof follows directly from Theorem 2.1 and Theorem 3.3.

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 3.2. Let \(\Gamma_{\Omega}\) be a non-commuting graph on \(D_{2n}\), where \(\Omega = \Omega_1 \cup \Omega_2\). Then

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4. The mean distance of the graph \(\Gamma_{\Omega}\)

Through this section we find the mean (average) distance of the graph \(\Gamma_{\Omega}\).

Lemma 4.1. In the graph \(\Gamma_{\Omega}\), where n is odd, the transmission of each vertex \(r^i\) is \(\sigma(r^i, \Gamma_{\Omega}) = 3n - 4\) for all \(1 \le i \le n - 1\) and the transmission of a vertex \(sr^i\) is \(\sigma(sr^i, \Gamma_{\Omega}) = 2n - 2\) for all \(1 \le i \le n\).

Proof. The vertex-set of the graph \(\Gamma_{\Omega}\) is \(V(\Gamma_{\Omega}) = \{r^i, sr^j : 1 \leq i < n, 1 \leq j \leq n\}\). Then \(|V(\Gamma_{\Omega})| = 2n - 1\), 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 to \(r^j\) 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,

\[\sigma(r^{i}, \Gamma_{\Omega}) = \sum_{\substack{1 \le j < n \\ i \ne i}} d(r^{i}, r^{j}) + \sum_{\substack{1 \le j \le n \\ i \ne i}} d(r^{i}, sr^{j}) = 2(n-2) + n = 3n - 4\]

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

\[\sigma(sr^{i}, \Gamma_{\Omega}) = \sum_{\substack{1 \le i, j \le n \\ i \ne j}} d(sr^{i}, sr^{j}) + \sum_{1 \le j < n} d(sr^{i}, r^{j}) = (n-1) + (n-1) = 2n - 2,\]

for all \[1 < i < n\].

Lemma 4.2. In the graph \(\Gamma_{\Omega}\), where n is even, the transmission of each vertex \(r^i\) is \(\sigma(r^i, \Gamma_{\Omega}) = 3n - 6\) for all \(1 \le i \le n - 1\) and the transmission of a vertex \(sr^i\) is \(\sigma(sr^i, \Gamma_{\Omega}) = 2n - 2\) for all \(1 \le i \le n\).

Proof. Let \(M=\{1,2,\ldots,n-1\}-\{n/2\}\). Then the vertex-set of the graph \(\Gamma_{\Omega}\), where n is even, is \(V(\Gamma_{\Omega})=\{r^i,sr^j:i\in M,1\leq j\leq n\}\). So, \(|V(\Gamma_{\Omega})|=2n-2\). 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,

\[\sigma(r^{i}, \Gamma_{\Omega}) = \sum_{\substack{j \in S \\ j \neq i}} d(r^{i}, r^{j}) + \sum_{1 \le j \le n} d(r^{i}, sr^{j}) = 2(n-3) + n = 3n - 6\]

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,

\[\sigma(sr^{i}, \Gamma_{\Omega}) = \sum_{\substack{1 \le j \le n \\ j \ne i}} d(sr^{i}, sr^{j}) + \sum_{j \in S} d(sr^{i}, r^{j}) = (n-2) + 2 + (n-2) = 2n - 2,\]

for all \[1 \leq i \leq n\].

Theorem 4.1. The mean distance of the graph \(\Gamma_{\Omega}\), where n is odd, is \(\mu(\Gamma_{\Omega}) = \frac{5n-4}{4n-2}\).

Proof. By Lemma 4.1, we see that the transmission of the graph \(\Gamma_{\Omega}\) is

\[\sigma(\Gamma_{\Omega}) = \sum_{i=1}^{n-1} \sigma(r^{i}, \Gamma_{\Omega}) + \sum_{i=1}^{n} \sigma(sr^{i}, \Gamma_{\Omega})\]
= \((n-1)(3n-4) + n(2n-2)\)
= \(5n^{2} - 9n + 4\).

Notice that \[|V(\Gamma_{\Omega})|=2n-1\]. Therefore, \(\mu(\Gamma_{\Omega})=\frac{\sigma(\Gamma_{\Omega})}{|V(\Gamma_{\Omega})|(|V(\Gamma_{\Omega})|-1)}=\frac{5n^2-9n+4}{(2n-1)(2n-2)}=\frac{5n-4}{4n-2}\). \(\square\)

Theorem 4.2. The mean distance of the graph \(\Gamma_{\Omega}\), where n is even, is \(\mu(\Gamma_{\Omega}) = \frac{5n^2 - 14n + 12}{(2n-2)(2n-3)}\).

Proof. By using Lemma 4.2, we can find the transmission of the graph \(\Gamma_{\Omega}\) which is

\[\sigma(\Gamma_{\Omega}) = \sum_{\substack{i=1\\i\neq n/2}}^{n-1} \sigma(r^{i}, \Gamma_{\Omega}) + \sum_{i=1}^{n} \sigma(sr^{i}, \Gamma_{\Omega})\]\[= (n-2)(3n-6) + n(2n-2)\]\[= 5n^{2} - 14n + 12.\]

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

Acknowledgement

The authors thank the anonymous referee for his/her careful reading.

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