Common Neighborhood Energy of Commuting Graphs of Finite Groups

Abstract

The commuting graph of a finite non-abelian group G with center $Z\left(G\right)$, denoted by ${\Gamma }_c\left(G\right)$, is a simple undirected graph whose vertex set is $G\setminus Z\left(G\right)$, and two distinct vertices x and y are adjacent if and only if $xy=yx$. Alwardi et al. (Bulletin, 2011, 36, 49-59) defined the common neighborhood matrix $CN\left(\mathcal{G}\right)$ and the common neighborhood energy $E_{cn}\left(\mathcal{G}\right)$ of a simple graph $\mathcal{G}$. A graph $\mathcal{G}$ is called CN-hyperenergetic if $E_{cn}\left(\mathcal{G}\right)>E_{cn}\left(K_n\right)$, where $n=\left|V\right(\mathcal{G}\left)\right|$ and $K_n$ denotes the complete graph on n vertices. Two graphs $\mathcal{G}$ and $\mathcal{H}$ with equal number of vertices are called CN-equienergetic if $E_{cn}\left(\mathcal{G}\right)=E_{cn}\left(\mathcal{H}\right)$. In this paper we compute the common neighborhood energy of ${\Gamma }_c\left(G\right)$ for several classes of finite non-abelian groups, including the class of groups such that the central quotient is isomorphic to group of symmetries of a regular polygon, and conclude that these graphs are not CN-hyperenergetic. We shall also obtain some pairs of finite non-abelian groups such that their commuting graphs are CN-equienergetic.

1. Introduction

Let $\mathcal{G}$ be a simple graph whose vertex set is $V\left(\mathcal{G}\right)=\{v_1,v_2,\cdots ,v_n\}$. The common neighborhood of two distinct vertices $v_i$ and $v_j$, denoted by $C(v_i,v_j)$, is the set of vertices adjacent to both $v_i$ and $v_j$ other than $v_i$ and $v_j$. The common neighborhood matrix of $\mathcal{G}$, denoted by $CN\left(\mathcal{G}\right)$, is a matrix of size n whose $(i,j)$th entry is 0 or $\left|C\right(v_i,v_j\left)\right|$ according as $i=j$ or $i\ne j$. The common neighborhood matrix is a symmetric matrix, hence all its eigenvalues are real. The common neighborhood eigenvalues are symmetric with respect to the origin for some special class of graphs. There is a nice relation between $CN\left(\mathcal{G}\right)$ and $A\left(\mathcal{G}\right)$, the adjacency matrix of $\mathcal{G}$. More precisely, if $i\ne j$ then the $(i,j)$th entry of $CN\left(\mathcal{G}\right)$ is same as the $(i,j)$th entry of $A{\left(\mathcal{G}\right)}^2$, which is the number of 2-walks between the vertices $v_i$ and $v_j$. Further, the $(i,i)$th entry of $CN\left(\mathcal{G}\right)$ is equal to the degree of $v_i$. Hence, $CN\left(\mathcal{G}\right)=A{\left(\mathcal{G}\right)}^2-D\left(\mathcal{G}\right)$, where $D\left(\mathcal{G}\right)$ is the degree matrix of $\mathcal{G}$. Let $CN-spec\left(\mathcal{G}\right)$ be the spectrum of $CN\left(\mathcal{G}\right)$. Then $CN-spec\left(\mathcal{G}\right)$ is the set of all the eigenvalues of $CN\left(\mathcal{G}\right)$ with multiplicities. If ${\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_k$ are the distinct eigenvalues of $CN\left(\mathcal{G}\right)$ with multiplicities $a_1,a_2,\cdots ,a_k$, respectively, then we write $CN-spec\left(\mathcal{G}\right)=\{{\alpha }_1^{a_1},{\alpha }_2^{a_2},\cdots ,{\alpha }_k^{a_k}\}$. The common neighborhood energy (abbreviated as CN-energy) of the graph $\mathcal{G}$ is given by

$$E_{cn}\left(\mathcal{G}\right)=\sum _{i=1}^k{a}_i\left|{\alpha }_i\right|.$$

The study of CN-energy of graphs was introduced by Alwardi et al. in [1]. Various properties of CN-energy of a graph can also be found in [1,2]. The motivation of studying $E_{cn}\left(\mathcal{G}\right)$ comes from the study of $E\left(\mathcal{G}\right)$, which is well-known as energy of $\mathcal{G}$, a notion introduced by Gutman [3]. Many results on $E\left(\mathcal{G}\right)$, including some bounds and chemical applications, can be found in [4,5,6,7,8,9,10,11,12,13,14,15]. It is worth recalling that $E\left(\mathcal{G}\right)$ is the sum of the absolute values of the eigenvalues of the adjacency matrix of $\mathcal{G}$. It is also interesting to note that $E\left(\mathcal{G}\right)$ can be obtained if $E_{cn}\left(\mathcal{G}\right)$ is known for some classes of graphs. For instance, $E\left(K_n\right)=E_{cn}\left(K_n\right)/(n-2)$ and $E\left(K_{m,n}\right)=\sqrt{E_{cn}\left(K_{m,n}\right)+2(n+n)}$, where $K_n$ is the complete graph on n vertices and $K_{m,n}$ is the complete bipartite graph on $(m+n)$ vertices. A graph $\mathcal{G}$ is called CN-hyperenergetic if $E_{cn}\left(\mathcal{G}\right)>E_{cn}\left(K_n\right)$, where $n=\left|V\right(\mathcal{G}\left)\right|$. It is still an open problem to produce a CN-hyperenergetic graph or to prove the non-existence of such graph (see [1] (Open problem 1)). In this paper we give an attempt to answer this problem by considering commuting graphs of finite groups.

The commuting graph of a finite non-abelian group G with center $Z\left(G\right)$ is a simple undirected graph whose vertex set is $G\setminus Z\left(G\right)$ and two vertices x and y are adjacent if and only if $xy=yx$. We write ${\Gamma }_c\left(G\right)$ to denote this graph. In [16,17,18,19,20,21,22,23], various aspects of ${\Gamma }_c\left(G\right)$ are studied. In Section 2 of this paper, we derive an expression for computing CN-energy of a particular class of graphs and list a few already known results. In Section 3, we compute CN-energy of commuting graph of certain metacyclic group, dihedral group (which is the group of symmetries of a regular polygon), quasidihedral group, generalized quarternion group, Hanaki group etc. We also consider some generalizations of dihedral group and generalized quarternion group. Two graphs $\mathcal{G}$ and $\mathcal{H}$ with equal number of vertices are called CN-equienergetic if $E_{cn}\left(\mathcal{G}\right)=E_{cn}\left(\mathcal{H}\right)$. In Section 3, we shall also obtain some pairs of finite non-abelian groups such that their commuting graphs are CN-equienergetic. As consequences of our results, in Section 4, we show that ${\Gamma }_c\left(G\right)$ for all G considered in Section 3 are not CN-hyperenergetic. We also identify some positive integers n such that ${\Gamma }_c\left(G\right)$ is not CN-hyperenergetic if G is an n-centralizer group. It is worth mentioning that CN-spectrums of ${\Gamma }_c\left(G\right)$ for certain classes of finite groups have been computed in [24] recently. However, the method adopted here, in computing CN-energy of ${\Gamma }_c\left(G\right)$ for various families of finite groups, is independent of $CN-spec\left({\Gamma }_c\left(G\right)\right)$.

Recall that an n-centralizer group G is a group such that $|Cent(G\left)\right|=n$, where $Cent\left(G\right)=\{C_G\left(w\right):w\in G\}$ and $C_G\left(w\right)=\{v\in G:vw=wv\}$ is the centralizer of w (see [25,26]). We also identify some $r\in {\mathbb{Q}}^{>0}$ such that ${\Gamma }_c\left(G\right)$ is not CN-hyperenergetic if $Pr\left(G\right)=r$. Also recall that the commutativity degree of G, denoted by $Pr\left(G\right)$, is the probability that a randomly chosen pair of elements of G commute.

Readers may review [27,28,29,30,31,32] for the background and various results regarding this notion. Further, we show that ${\Gamma }_c\left(G\right)$ is not CN-hyperenergetic if ${\Gamma }_c\left(G\right)$ is not planar or toroidal. Note that a graph is planar or toroidal according as its genus is zero or one respectively. Finally, we conclude the paper with a few conjectures.

2. A Useful Formula and Prerequisites

We write $\mathcal{G}={\mathcal{G}}_1\bigsqcup {\mathcal{G}}_2$ to denote that $\mathcal{G}$ has two components namely ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$. Also, $l{K}_m$ denotes the disjoint union of l copies of the complete graph $K_m$ on m vertices. We begin this section with the following two key results of Alwardi et al. [1].

Theorem 1 

([1] Proposition 2.4). If $\mathcal{G}={\mathcal{G}}_1\bigsqcup {\mathcal{G}}_2\bigsqcup \cdots \bigsqcup {\mathcal{G}}_m$ then $E_{cn}\left(\mathcal{G}\right)=\underset{i=1}{\sum ^m}{E}_{cn}\left({\mathcal{G}}_i\right).$

Lemma 1 

([1] Example 2.1). If $K_n$ denotes the complete graph on n vertices then

$$E_{cn}\left(K_n\right)=2(n-1)(n-2).$$

Now we derive a formula for CN-energy of graphs which are disjoint unions of some complete graphs. The following theorem is very useful in order to compute CN-energy of commuting graphs of finite groups.

Theorem 2.

Let $\mathcal{G}=l_1{K}_{m_1}\bigsqcup l_2{K}_{m_2}\bigsqcup \cdots \bigsqcup l_k{K}_{m_k}$, where $l_i{K}_{m_i}$ denotes the disjoint union of $l_i$ copies of the complete graphs $K_{m_i}$ on $m_i$ vertices for $1\le i\le k$. Then

$$E_{cn}\left(\mathcal{G}\right)=2\sum _{i=1}^k{l}_i(m_i-1)(m_i-2).$$

Proof. 

By Theorem 1 we have

$$E_{cn}\left(\mathcal{G}\right)=\sum _{i=1}^k{l}_i{E}_{cn}\left(K_{m_i}\right).$$

Therefore, the result follows from Lemma 1. □

We conclude this section with the following useful results from [17,18].

Lemma 2.

Let G be a finite group with center $Z\left(G\right)$. If $\frac{G}{Z\left(G\right)}$ is isomorphic to

1.

The Suzuki group $Sz\left(2\right)$, presented by $\langle u,v:u^5=v^4=1,v^{-1}uv=u^2\rangle$, then ${\Gamma }_c\left(G\right)=5K_{3\left|Z\right(G\left)\right|}\bigsqcup K_{4\left|Z\right(G\left)\right|}$.

2.

${\mathbb{Z}}_p\times {\mathbb{Z}}_p$, for any prime p, then ${\Gamma }_c\left(G\right)=(p+1)K_{(p-1)\left|Z\right(G\left)\right|}$.

3.

The dihedral group $D_{2m}$$(m\ge 2)$, presented by $\langle u,v:u^m=v^2=1,vu{v}^{-1}=u^{-1}\rangle$, then ${\Gamma }_c\left(G\right)=K_{(m-1)\left|Z\right(G\left)\right|}\bigsqcup m{K}_{\left|Z\right(G\left)\right|}$.

Lemma 3.

Let G be a non-abelian group. If G is isomorphic to

1.

A group of order $pq$, where p and q are primes with $p\mid (q-1)$, then ${\Gamma }_c\left(G\right)=K_{q-1}\bigsqcup q{K}_{p-1}$.

2.

The quasidihedral group $Q{D}_{2^n}$$(n\ge 4)$, presented by $\langle u,v:u^{2^{n-1}}=v^2=1,vu{v}^{-1}=u^{2^{n-2}-1}\rangle$, then ${\Gamma }_c\left(G\right)=K_{2^{n-1}-2}\bigsqcup 2^{n-2}{K}_2$.

3.

$PSL(2,2^k)$, the projective special linear group for $k\ge 2$, then ${\Gamma }_c\left(G\right)=2^{k-1}(2^k-1)K_{2^k}\bigsqcup (2^k+1)K_{2^k-1}\bigsqcup 2^{k-1}(2^k+1)K_{2^k-2}$.

4.

$GL(2,q)$, the general linear group where $q=p^n>2$ and p is a prime, then ${\Gamma }_c\left(G\right)=\frac{q(q-1)}{2}{K}_{q^2-q}\bigsqcup \frac{q(q+1)}{2}{K}_{q^2-3q+2}\bigsqcup (q+1)K_{q^2-2q+1}$.

Lemma 4.

Let G be a non-abelian group. If G is isomorphic to

1.

The Hanaki group $A(n,\sigma )$$(n\ge 2)$ of order $2^{2n}$ given by

$$\left\{U(x,y)=\left[\begin{array}{ccc}1& 0& 0\\ x& 1& 0\\ y& \sigma \left(x\right)& 1\end{array}\right]:x,y\in F\right\},$$

under matrix multiplication where $F=GF\left(2^n\right)$ and $\sigma \in Aut\left(F\right)$ given by $\sigma \left(u\right)=u^2$, then ${\Gamma }_c\left(G\right)=(2^n-1)K_{2^n}$.

2.

The Hanaki group $A(n,p)$ of order $p^{3n}$ given by

$$\left\{V(x,y,z)=\left[\begin{array}{ccc}1& 0& 0\\ x& 1& 0\\ y& z& 1\end{array}\right]:x,y,z\in F\right\},$$

under matrix multiplication where $F=GF\left(p^n\right)$ and p is a prime, then ${\Gamma }_c\left(G\right)=(p^n+1)K_{p^{2n}-p^n}$.

3. CN-Energy of Commuting Graphs

In this section, we compute $E_{cn}\left({\Gamma }_c\left(G\right)\right)$ for several classes of finite non-abelian groups.

Theorem 3.

Let G be a finite group with center $Z\left(G\right)$. If $\frac{G}{Z\left(G\right)}$ is isomorphic to

1.

The Suzuki group $Sz\left(2\right)$, then

$$E_{cn}\left({\Gamma }_c\left(G\right)\right)={2\left(61\right|Z\left(G\right)|}^2-57\left|Z\left(G\right)\right|+12).$$

2.

${\mathbb{Z}}_p\times {\mathbb{Z}}_p$, then

$$E_{cn}\left({\Gamma }_c\left(G\right)\right)=2(p+1)\left((p-1)\right|Z\left(G\right)|-1)\left((p-1)\right|Z\left(G\right)|-2).$$

3.

The dihedral group $D_{2m}$$(m\ge 2)$, then

$$E_{cn}\left({\Gamma }_c\left(G\right)\right)=2((m^2-m+1)|Z\left(G\right){|}^2-(6m-3)|Z\left(G\right)|+2m+2).$$

Proof. 

By Lemma 2 and Theorem 2 we have

$$E_{cn}\left({\Gamma }_c\left(G\right)\right)=\left\{\begin{array}{cc}2\left(4\right|Z\left(G\right)|-1)\left(4\right|Z\left(G\right)|-2)+10\left(3\right|Z\left(G\right)|-1)\left(3\right|Z\left(G\right)|-2),\hfill & \\ \mathrm{if}\frac{G}{Z\left(G\right)}\cong Sz\left(2\right)\hfill \\ 2(p+1)\left(\right(p-1\left)\right|Z\left(G\right)|-1)\left(\right(p-1\left)\right|Z\left(G\right)|-2),\hfill \\ \mathrm{if}\frac{G}{Z\left(G\right)}\cong {\mathbb{Z}}_p\times {\mathbb{Z}}_p\hfill \\ 2\left(\right(m-1\left)\right|Z\left(G\right)|-1)\left(\right(m-1\left)\right|Z\left(G\right)|-2)\hfill & \\ +2m\left(\right|Z\left(G\right)|-1)\left(\right|Z\left(G\right)|-2),\mathrm{if}\frac{G}{Z\left(G\right)}\cong D_{2m}.\hfill \end{array}\right.$$

Hence, the result follows on simplification. □

We have the following two corollaries of Theorem 3.

Corollary 1.

Let G be isomorphic to one of the following groups

1.

${\mathbb{Z}}_2\times Q_8,$

2.

${\mathbb{Z}}_2\times D_8,$

3.

${\mathbb{Z}}_4⋊{\mathbb{Z}}_4=\langle u,v:u^4=v^4=1,vu{v}^{-1}=u^{-1}\rangle ,$

4.

${\mathcal{M}}_{16}=\langle u,v:u^8=v^2=1,vuv=u^5\rangle ,$

5.

$SG(16,3)=\langle u,v:u^4=v^4=1,uv=v^{-1}{u}^{-1},u{v}^{-1}=v{u}^{-1}\rangle ,$

6.

$D_8*{\mathbb{Z}}_4=\langle u,v,w:u^4=v^2=w^2=1,uv=vu,uw=wu,vw=u^{2}wv\rangle .$

Then $E_{cn}\left({\Gamma }_c\left(G\right)\right)=36.$

Proof. 

If G is isomorphic to one of the above listed group then it is of order 16. Therefore, $\left|Z\right(G\left)\right|=4$ and so $\frac{G}{Z\left(G\right)}\cong {\mathbb{Z}}_2\times {\mathbb{Z}}_2$. Hence, putting $p=2$ in Theorem 3 (2) we get the required result. □

Corollary 2.

Let G be a non-abelian group.

1.

If G is of order $p^3$, for any prime p, then

$$E_{cn}\left({\Gamma }_c\left(G\right)\right)=2(p+1)(p^2-p-1)(p^2-p-2).$$

2.

If G is the metacyclic group $M_{2mn}$$(m\ge 3)$, presented by $\langle u,v:u^m=v^{2n}=1,vu{v}^{-1}=u^{-1}\rangle$, then

$$E_{cn}\left({\Gamma }_c\left(G\right)\right)=\left\{\begin{array}{cc}2((m^2-m+1)n^2-(6m-3)n+2m+2),\hfill & \mathrm{if}m\mathrm{is}\mathrm{odd}\hfill \\ 2((m^2-2m+4)n^2-(6m-6)n+m+2),\hfill & \mathrm{if}m\mathrm{is}\mathrm{even}.\hfill \end{array}\right.$$

3.

If G is the dihedral group $D_{2m}$$(m\ge 3)$, then

$$E_{cn}\left({\Gamma }_c\left(G\right)\right)=\left\{\begin{array}{cc}2(m-2)(m-3),\hfill & \mathrm{if}m\mathrm{is}\mathrm{odd}\hfill \\ 2(m-3)(m-4),\hfill & \mathrm{if}m\mathrm{is}\mathrm{even}.\hfill \end{array}\right.$$

4.

If G is the generalized quaternion group $Q_{4n}$$(n\ge 2)$, presented by $\langle u,v:v^{2n}=1,u^2=v^n,uv{u}^{-1}=v^{-1}\rangle$, then

$$E_{cn}\left({\Gamma }_c\left(G\right)\right)=2(2n-3)(2n-4).$$

Proof. 

If G is of order $p^3$ then $\left|Z\right(G\left)\right|=p$ and $\frac{G}{Z\left(G\right)}\cong {\mathbb{Z}}_p\times {\mathbb{Z}}_p$. Hence the result follows from Theorem 3 (2).

(2) We have

$$|Z\left(M_{2mn}\right)|=\left\{\begin{array}{cc}n,\hfill & \mathrm{if}m\mathrm{is}\mathrm{odd}\hfill \\ 2n,\hfill & \mathrm{if}m\mathrm{is}\mathrm{even}\hfill \end{array}\right.\mathrm{and}\frac{M_{2mn}}{Z\left(M_{2mn}\right)}\cong \left\{\begin{array}{cc}{D}_{2m},\hfill & \mathrm{if}m\mathrm{is}\mathrm{odd}\hfill \\ D_m,\hfill & \mathrm{if}m\mathrm{is}\mathrm{even}.\hfill \end{array}\right.$$

Hence, the result follows from Theorem 3 (3).

(3) Follows from part (2), considering $n=1$.

(4) Follows from Theorem 3 (3), since $\left|Z\right(Q_{4n}\left)\right|=2$ and $\frac{Q_{4n}}{Z\left(Q_{4n}\right)}\cong D_{2n}$. □

In the following theorems we compute $E_{cn}\left({\Gamma }_c\left(G\right)\right)$ for more families of groups.

Theorem 4.

Let G be a non-abelian group.

1.

If G is of order $pq$, where p and q are primes with $p\mid (q-1)$, then

$$E_{cn}\left({\Gamma }_c\left(G\right)\right)=2(q^2+p^{2}q-5pq+q+6).$$

2.

If G is the quasidihedral group $Q{D}_{2^n}$$(n\ge 4)$, then

$$E_{cn}\left({\Gamma }_c\left(G\right)\right)=2(2^{n-1}-3)(2^{n-1}-4).$$

3.

If $G=PSL(2,2^k)$ then

$$E_{cn}\left({\Gamma }_c\left(G\right)\right)=2^{4k+1}-4·2^{3k+1}+2^{2k+1}+6·2^{k+1}+12.$$

4.

If $G=GL(2,q)$ then

$$E_{cn}\left({\Gamma }_c\left(G\right)\right)=2q^6-6q^5-2q^4+10q^3+6q^2+2q.$$

Proof. 

(1) If G is of order $pq$ then, by Lemma 3 (1) and Theorem 2, we have

$$E_{cn}\left({\Gamma }_c\left(G\right)\right)=2((q-2)(q-3)+q(p-2)(p-3)).$$

This gives the required result on simplification.

(2) Follows from Lemma 3 (2) and Theorem 2.

(3) By Lemma 3 (3) and Theorem 2 we have

$$\begin{array}{ccc}& & \frac{E_{cn}\left({\Gamma }_c\left(G\right)\right)}{2}=(2^k+1)(2^k-2)(2^k-3)+2^{k-1}(2^k+1)(2^k-3)(2^k-4)\hfill \\ & & +2^{k-1}(2^k-1)(2^k-1)(2^k-2),\hfill \end{array}$$

which gives the required result.

(4) By Lemma 3 (4) and Theorem 2 we have

$$\begin{array}{ccc}& & E_{cn}\left({\Gamma }_c\left(G\right)\right)=q(q+1)(q^2-3q+1)(q^2-3q)+q(q-1)(q^2-q-1)(q^2-q-2)\hfill \\ & & +2(q+1)(q^2-2q)(q^2-2q-1),\hfill \end{array}$$

which gives the required result on simplification. □

Theorem 5.

Let G be a non-abelian group.

1.

If G is the Hanaki group $A(n,\sigma )$ then

$$E_{cn}\left({\Gamma }_c\left(G\right)\right)=2{(2^n-1)}^2(2^n-2).$$

2.

If G is the Hanaki group $A(n,p)$ then

$$E_{cn}\left({\Gamma }_c\left(G\right)\right)=2(p^n+1)(p^{2n}-p^n-1)(p^{2n}-p^n-2).$$

Proof. 

The result follows from Lemma 4 and Theorem 2. □

Note that all the groups considered above are abelian centralizer group (in short, AC-group). Now we present a result on $E_{cn}\left({\Gamma }_c\left(G\right)\right)$ if G is a finite AC-group.

Theorem 6.

Consider that an AC-group G has distinct centralizers $X_1,\cdots ,X_n$ of non-central elements of G. Then $E_{cn}\left({\Gamma }_c\left(G\right)\right)=2\underset{i=1}{\sum ^n}\left(\right|X_i|-|Z\left(G\right)|-1)\left(\right|X_i|-|Z\left(G\right)|-2).$

Proof. 

We have ${\Gamma }_c\left(G\right)=\stackrel{n}{\bigsqcup _{i=1}}{K}_{|X_i|-|Z\left(G\right)|}$, by [17] (Lemma 1). Therefore, by Theorem 2, the result follows. □

Corollary 3.

Let K be a finite abelian group and H be a finite non-abelian AC-group. If $G\cong H\times K$ then

$$\begin{array}{c}\hfill E_{cn}\left({\Gamma }_c\left(G\right)\right)=2\underset{i=1}{\sum ^n}\left(\right|Y_i\left|\right|K|-\left|Z\left(H\right)\right|\left|K\right|-1\left)\right(|Y_i\left|\right|K|-|Z\left(H\right)\left|\right|K|-2),\end{array}$$

where $Cent\left(H\right)=\{H,Y_1,\cdots ,Y_n\}$.

Proof. 

Clearly $Z(H\times K)=Z\left(H\right)\times K$ and $Cent(H\times K)=\{H\times K,Y_1\times K,Y_2\times K,\cdots ,Y_n\times K\}$. Hence, $H\times K$ is an AC-group and so, by Theorem 6, the result follows. □

We shall conclude this section by obtaining some pairs of finite non-abelian groups such that their commuting graphs are CN-equienergetic.

Proposition 1.

The commuting graphs of $D_{4k}$ and $Q_{4k}$ for $k\ge 2$ are CN-equienergetic.

Proof. 

The result follows from parts (3) and (4) of Corollary 2. □

Using Corollary 2 (parts (3) and (4)) and Theorem 4 (2) we also have the following result.

Proposition 2.

The commuting graphs of $D_{2^k}$, $Q_{2^k}$ and $Q{D}_{2^k}$ for $k\ge 4$ are pairwise CN-equienergetic.

4. Some Consequences

In this section we derive some consequences of the results obtained in Section 3.

Theorem 7.

Let G be a finite group with center $Z\left(G\right)$. If $\frac{G}{Z\left(G\right)}$ is isomorphic to $Sz\left(2\right),{\mathbb{Z}}_p\times {\mathbb{Z}}_p$ or $D_{2m}$ (where p is any prime and $m\ge 2$) then ${\Gamma }_c\left(G\right)$ is not CN-hyperenergetic.

Proof. 

If $\frac{G}{Z\left(G\right)}\cong Sz\left(2\right)$ then, by Theorem 3 (1), we have

$$E_{cn}\left({\Gamma }_c\left(G\right)\right)={2\left(61\right|Z\left(G\right)|}^2-57\left|Z\left(G\right)\right|+12).$$

Since $\left|V\right({\Gamma }_c\left(G\right)\left)\right|=19\left|Z\left(G\right)\right|$, by Lemma 1 we have

$$E_{cn}\left(K_{19\left|Z\right(G\left)\right|}\right)=2\left(19\right|Z\left(G\right)|-1)\left(19\right|Z\left(G\right)|-2)=2\left(361\right|Z\left(G\right){|}^2-57|Z\left(G\right)|+2).$$

Clearly, ${361\left|Z\left(G\right)\right|}^2+2>61{\left|Z\left(G\right)\right|}^2+12$ which gives $E_{cn}\left(K_{19\left|Z\right(G\left)\right|}\right)>E_{cn}\left({\Gamma }_c\left(G\right)\right)$.

If $\frac{G}{Z\left(G\right)}\cong {\mathbb{Z}}_p\times {\mathbb{Z}}_p$ then, by Theorem 3 (2), we have

$$E_{cn}\left({\Gamma }_c\left(G\right)\right)=2(p+1)\left((p-1)\right|Z\left(G\right)|-1)\left((p-1)\right|Z\left(G\right)|-2).$$

Since $|V\left({\Gamma }_c\left(G\right)\right)|=(p^2-1)\left|Z\left(G\right)\right|$, by Lemma 1 we have

$$E_{cn}\left(K_{(p^2-1)\left|Z\left(G\right)\right|}\right)=2((p^2-1)|Z\left(G\right)|-1)((p^2-1)|Z\left(G\right)|-2).$$

Clearly

$$\begin{array}{cc}\hfill & ((p^2-1)|Z\left(G\right)|-1)((p^2-1)|Z\left(G\right)|-2)\hfill \\ \hfill & >((p^2-1)|Z\left(G\right)|-(p+1)\left)\right((p^2-1)|Z\left(G\right)|-2(p+1))\hfill \\ \hfill & >(p+1)\left(\right(p-1\left)\right|Z\left(G\right)|-1)\left(\right(p-1\left)\right|Z\left(G\right)|-2.\hfill \end{array}$$

Thus $E_{cn}\left(K_{(p^2-1)\left|Z\left(G\right)\right|}\right)>E_{cn}\left({\Gamma }_c\left(G\right)\right)$.

If $\frac{G}{Z\left(G\right)}\cong D_{2m}$ then we have

$$E_{cn}\left({\Gamma }_c\left(G\right)\right)=2((m^2-m+1)|Z\left(G\right){|}^2-(6m-3)|Z\left(G\right)|+2m+2),$$

by Theorem 3 (3). Since $\left|V\right({\Gamma }_c\left(G\right)\left)\right|=(2m-1)\left|Z\left(G\right)\right|$, by Lemma 1 we have

$$\begin{array}{cc}\hfill E_{cn}\left(K_{(2m-1)\left|Z\right(G\left)\right|}\right)& =2\left(2m\right|Z\left(G\right)|-|Z\left(G\right)|-1)\left(2m\right|Z\left(G\right)|-|Z\left(G\right)|-2)\hfill \\ \hfill & =2\left((4m^2-4m+1)\right|Z\left(G\right){|}^2-(6m-3)\left|Z\left(G\right)\right|+2).\hfill \end{array}$$

Clearly $(4m^2-4m+1)|Z\left(G\right){|}^2>(m^2-m+1){\left|Z\left(G\right)\right|}^2+2m$. Therefore,

$E_{cn}\left(K_{(p^2-1)\left|Z\left(G\right)\right|}\right)$$>E_{cn}\left({\Gamma }_c\left(G\right)\right)$. This completes the proof. □

We have the following two corollaries.

Corollary 4.

If G is isomorphic to one of the groups listed in Corollary 1, then ${\Gamma }_c\left(G\right)$ is not CN-hyperenergetic.

Proof. 

Since $\frac{G}{Z\left(G\right)}$ is isomorphic to ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$, the result follows from Theorem 7 considering $p=2$. □

Corollary 5.

Let G be a non-abelian group. If G is isomorphic to $M_{2mn},D_{2m},Q_{4n}$ or a group of order $p^3$ then ${\Gamma }_c\left(G\right)$ is not CN-hyperenergetic.

Proof. 

If G is isomorphic to $M_{2mn},D_{2m}$ or $Q_{4n}$ then $\frac{G}{Z\left(G\right)}$ is isomorphic to some dihedral groups. If G is isomorphic to a group of order $p^3$ then $\frac{G}{Z\left(G\right)}$ is isomorphic to ${\mathbb{Z}}_p\times {\mathbb{Z}}_p$. Hence, by Theorem 7, the result follows. □

We have the following results regarding commuting graphs of finite n-centralizer groups.

Theorem 8.

If G is a finite 4-centralizer group then ${\Gamma }_c\left(G\right)$ is not CN-hyperenergetic.

Proof. 

We have $\frac{G}{Z\left(G\right)}\cong {\mathbb{Z}}_2\times {\mathbb{Z}}_2$, by [25] (Theorem 2). Hence, using Theorem 7 for $p=2$, the result follows. □

Theorem 9.

Let G be a finite $(p+2)$-centralizer p-group. Then ${\Gamma }_c\left(G\right)$ is not CN-hyperenergetic.

Proof. 

We have $\frac{G}{Z\left(G\right)}\cong {\mathbb{Z}}_p\times {\mathbb{Z}}_p$, by [33] (Lemma 2.7). Hence, by Theorem 7, the result follows. □

Theorem 10.

If G is a finite 5-centralizer group then ${\Gamma }_c\left(G\right)$ is not CN-hyperenergetic.

Proof. 

We have $\frac{G}{Z\left(G\right)}\cong {\mathbb{Z}}_3\times {\mathbb{Z}}_3$ or $D_6$, by [25] (Theorem 4). Hence, by Theorem 7, the result follows. □

As a corollary to Theorems 8 and 10 we have the following result.

Corollary 6.

Let G be a finite non-abelian group and $\{x_1,x_2,\cdots ,x_r\}$ be a set of pairwise non-commuting elements of G having maximal size. Then ${\Gamma }_c\left(G\right)$ is not CN-hyperenergetic if $r=3,4$.

Proof. 

By [34] (Lemma 2.4), we have that G is a 4-centralizer or a 5-centralizer group according as $r=3$ or 4. Hence the result follows from Theorems 8 and 10. □

Theorem 11.

Let G be a non-abelian group. If G is isomorphic to $Q{D}_{2^n}$, $PSL(2,2^k)$, $A(n,\sigma )$, $GL(2,q)$, $A(n,p)$ or a group of order $pr$, where p and r are primes with $p\mid (r-1)$ and $q=p^m>2$, then ${\Gamma }_c\left(G\right)$ is not CN-hyperenergetic.

Proof. 

If G is isomorphic to $Q{D}_{2^n}$ then, by Theorem 4, we have $E_{cn}\left({\Gamma }_c\left(G\right)\right)=2(2^{n-1}-3)(2^{n-1}-4).$ Since $|V\left({\Gamma }_c\left(G\right)\right)|=2^n-2$, by Lemma 1 we have

$$E_{cn}\left(K_{2^n-2}\right)=2(2^n-3)(2^n-4).$$

Clearly, $(2^n-3)(2^n-4)>(2^{n-1}-3)(2^{n-1}-4)$. Hence, $E_{cn}\left(K_{2^n-2}\right)>E_{cn}\left({\Gamma }_c\left(G\right)\right)$.

If G is isomorphic to $PSL(2,2^k)$ then, by Theorem 4 (3), we have

$$E_{cn}\left({\Gamma }_c\left(G\right)\right)=2^{4k+1}-4·2^{3k+1}+2^{2k+1}+6·2^{k+1}+12.$$

Since $|V\left({\Gamma }_c\left(G\right)\right)|=2^k(2^{2k}-1)-1=2^{3k}-2^k-1$, by Lemma 1 we have

$$\begin{array}{cc}\hfill E_{cn}\left(K_{2^{3k}-2^k-1}\right)& =2(2^{3k}-2^k-1)(2^{3k}-2^k-3)\hfill \\ \hfill & =2^{6k+1}-2·2^{4k+1}-3·2^{3k+1}+2^{2k+1}+5·2^{k+1}+12.\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill E_{cn}\left(K_{2^{3k}-2^k-1}\right)-E_{cn}\left({\Gamma }_c\left(G\right)\right)& =2^{6k+1}-3·2^{4k+1}+2^{3k+1}-2^{k+1}\hfill \\ \hfill & =2^{4k+1}(2^{2k}-3)+2^{k+1}(2^{2k}-1).\hfill \end{array}$$

Since $2^{2k}-3>0$ and $2^{2k}-1>0$ we have $E_{cn}\left(K_{2^{3k}-2^k-1}\right)-E_{cn}\left({\Gamma }_c\left(G\right)\right)$ is positive. Hence, the result follows.

If G is isomorphic to $GL(2,q)$ then, by Theorem 4 (4), we have

$$E_{cn}\left({\Gamma }_c\left(G\right)\right)=2q^6-6q^5-2q^4+10q^3+6q^2+2q.$$

Since $|V\left({\Gamma }_c\left(G\right)\right)|=(q^2-1)(q^2-q)-(q-1)=q^4-q^3-q^2+1$, by Lemma 1 we have

$$E_{cn}\left(K_{q^4-q^3-q^2+1}\right)=2(q^4-q^3-q^2)(q^4-q^3-q^2-1)=2q^8-4q^7-2q^6+4q^5+2q^3+2q^2.$$

Therefore,

$$\begin{array}{cc}\hfill E_{cn}\left(K_{q^4-q^3-q^2+1}\right)-E_{cn}\left({\Gamma }_c\left(G\right)\right)& =2q^8-4q^7-4q^6+10q^5+2q^4-8q^3-4q^2-2q\hfill \\ \hfill & =2q^6(q^2-2q-2)+2q^2(5q^3-4q-2)+2q(q^3-2).\hfill \end{array}$$

We have $q^2-2q-2=q(q-2)-2>0$, $5q^3-4q-2=q(5q^2-4)-2>0$ and $q^3-2>0$ since $q=p^m>2$ for some prime p. Therefore, $E_{cn}\left(K_{q^4-q^3-q^2+1}\right)-E_{cn}\left({\Gamma }_c\left(G\right)\right)$ is positive and hence the result follows.

If G is isomorphic to $A(n,\sigma )$ then, by Theorem 5 (1), we have $E_{cn}\left({\Gamma }_c\left(G\right)\right)=2{(2^n-1)}^2(2^n-2).$ Since $|V\left({\Gamma }_c\left(G\right)\right)|=2^n(2^n-1)=2^{2n}-2^n$, by Lemma 1 we have

$$E_{cn}\left(K_{2^{2n}-2^n}\right)=2(2^{2n}-2^n-1)(2^{2n}-2^n-2).$$

Clearly, $2^{2n}-2^n-1>2^{2n}-2·2^n-1={(2^n-1)}^2$ and $2^{2n}-2^n-2>2^n-2$. Therefore, $E_{cn}\left(K_{2^{2n}-2^n}\right)>E_{cn}\left({\Gamma }_c\left(G\right)\right)$.

If $G\cong A(n,p)$ then, by Theorem 5 (2), we have $E_{cn}\left({\Gamma }_c\left(G\right)\right)=2(p^n+1)(p^{2n}-p^n-1)(p^{2n}-p^n-2).$ Since $|V\left({\Gamma }_c\left(G\right)\right)|=(p^n+1)(p^{2n}-p^n)$, by Lemma 1 we have

$$E_{cn}\left(K_{(p^n+1)(p^{2n}-p^n)}\right)=2((p^n+1)(p^{2n}-p^n)-1)((p^n+1)(p^{2n}-p^n)-2).$$

We have

$$\begin{array}{cc}\hfill & (p^n+1)(p^{2n}-p^n-1)(p^{2n}-p^n-2)\hfill \\ \hfill & <(p^n+1)(p^{2n}-p^n-1)(p^n+1)(p^{2n}-p^n-2)\hfill \\ \hfill & =((p^n+1)(p^{2n}-p^n)-(p^n+1))((p^n+1)(p^{2n}-p^n)-2(p^n+1))\hfill \\ \hfill & <((p^n+1)(p^{2n}-p^n)-1)((p^n+1)(p^{2n}-p^n)-2).\hfill \end{array}$$

Hence, $E_{cn}\left({\Gamma }_c\left(G\right)\right)<E_{cn}\left(K_{(p^n+1)(p^{2n}-p^n)}\right)$.

If G is isomorphic to a non-abelian group of order $pr$ then, by Theorem 4 (1), we have

$$E_{cn}\left({\Gamma }_c\left(G\right)\right)=2(r^2+p^{2}r-5pr+r+6).$$

Since $\left|V\right({\Gamma }_c\left(G\right)\left)\right|=pr-1$, by Lemma 1 we have

$$E_{cn}\left(K_{pr-1}\right)=2(pr-2)(pr-3)=2(p^2{r}^2-5pr+6).$$

Since $r+1\le 2(r-1)<p^2(r-1)$ we have $r^2+p^{2}r+r<p^2{r}^2$. Hence, $E_{cn}\left(K_{pr-1}\right)>E_{cn}\left({\Gamma }_c\left(G\right)\right)$. This completes the proof. □

It is already mentioned that $Pr\left(G\right)$, the commutativity degree of a group G, is the probability that a randomly chosen pair of elements of G commute. Therefore, it measures the abelianness of a group. For any finite group G, its commutativity degree can be computed using the formula

$$Pr\left(G\right)=\frac{1}{{\left|G\right|}^2}\sum _{w\in G}\left|C_G\left(w\right)\right|\mathrm{or}Pr\left(G\right)=\frac{k\left(G\right)}{\left|G\right|},$$

where $k\left(G\right)$ is the number of conjugacy classes in G. In finite group theory, it is an interesting problem to find all the rational numbers $r\in (0,1]$ such that $Pr\left(G\right)=r$ for some finite group G. Over the decades, many values of such r have obtained and characterized finite groups such that $Pr\left(G\right)=r$. In the following theorem we list some values of r such that ${\Gamma }_c\left(G\right)$ is not CN-hyperenergetic if $Pr\left(G\right)=r$.

Theorem 12.

If $Pr\left(G\right)\in \{\frac{5}{14},\frac{2}{5},\frac{11}{27},\frac{7}{16},\frac{1}{2},\frac{5}{8}\}$ then ${\Gamma }_c\left(G\right)$ is not CN-hyperenergetic.

Proof. 

If $Pr\left(G\right)\in \{\frac{5}{14},\frac{2}{5},\frac{11}{27},\frac{7}{16},\frac{1}{2},\frac{5}{8}\}$ then $\frac{G}{Z\left(G\right)}$ is isomorphic to the groups in $\{D_{14},D_{10},D_8,$$D_6,{\mathbb{Z}}_2\times {\mathbb{Z}}_2,{\mathbb{Z}}_3\times {\mathbb{Z}}_3\}$ (by [35] (p. 246) and [36] (p. 451)). Hence, the result follows from Theorem 7. □

Theorem 13.

Let G be a finite group and $Pr\left(G\right)=\frac{p^2+p-1}{p^3}$, where p is the smallest prime divisor of $\left|G\right|$. Then ${\Gamma }_c\left(G\right)$ is not CN-hyperenergetic.

Proof. 

We have $\frac{G}{Z\left(G\right)}\cong {\mathbb{Z}}_p\times {\mathbb{Z}}_p$, by [37] (Theorem 3). Hence the result follows from Theorem 7. □

Theorem 14.

If G is a finite non-solvable group and $Pr\left(G\right)=\frac{1}{12}$ then ${\Gamma }_c\left(G\right)$ is not CN-hyperenergetic.

Proof. 

We have $G\cong A_5\times K$ for some abelian group K, by [27] (Proposition 3.3.7). It can be seen that ${\Gamma }_c\left(G\right)=5K_{3\left|K\right|}\bigsqcup 10K_{2\left|K\right|}\bigsqcup 6K_{4\left|K\right|}$. Therefore, by Theorem 2, we have

$$\begin{array}{cc}\hfill E_{cn}\left({\Gamma }_c\left(G\right)\right)& =2\left(5\right(3\left|K\right|-1\left)\right(3\left|K\right|-2)+10(2\left|K\right|-1\left)\right(2\left|K\right|-2\left)\right)+6\left(4\right|K|-1)\left(4\right|K|-2))\hfill \\ \hfill & =2\left(181\right|K{|}^2-177\left|K\right|+42).\hfill \end{array}$$

Additionally, by Lemma 1, we have $E_{cn}\left(K_{59\left|K\right|}\right)={2\left(3481\right|K|}^2-177\left|K\right|+2)$. Therefore

$$E_{cn}\left(K_{59\left|K\right|}\right)-E_{cn}\left({\Gamma }_c\left(G\right)\right)={2\left(3300\right|K|}^2-40)>0.$$

This completes the proof. □

The following three theorems show that ${\Gamma }_c\left(G\right)$ is not CN-hyperenergetic if ${\Gamma }_c\left(G\right)$ is planar/toroidal or the complement of ${\Gamma }_c\left(G\right)$ is planar.

Theorem 15.

Let G be a finite non-abelian group. If ${\Gamma }_c\left(G\right)$ is planar then ${\Gamma }_c\left(G\right)$ is not CN-hyperenergetic.

Proof. 

If $G\cong D_{12}$, $D_{10},D_8,D_6$, $Q_8$ or $Q_{12}$ then, by Corollary 5, we have that ${\Gamma }_c\left(G\right)$ is not CN-hyperenergetic.

If G is isomorphic to one of the groups listed in Corollary 1 then, by Corollary 4, it follows that ${\Gamma }_c\left(G\right)$ is not CN-hyperenergetic. If $G\cong A_4$ then it can be seen that ${\Gamma }_c\left(G\right)=K_3\bigsqcup 4K_2$. Using Theorem 2, we have $E_{cn}\left({\Gamma }_c\left(G\right)\right)=4$. Also, by Lemma 1, we have $E_{cn}\left(K_{11}\right)=180$. Therefore, ${\Gamma }_c\left(G\right)$ is not CN-hyperenergetic. If $G\cong Sz\left(2\right)$ then $\frac{G}{Z\left(G\right)}\cong Sz\left(2\right)$. Therefore, by Theorem 7, it follows that ${\Gamma }_c\left(G\right)$ is not CN-hyperenergetic. If $G\cong SL(2,3)$ then it can be seen that ${\Gamma }_c\left(G\right)=3K_2\bigsqcup 4K_4$. Therefore, by Theorem 2, we have $E_{cn}\left({\Gamma }_c\left(G\right)\right)=48$. Also, by Lemma 1, we have $E_{cn}\left(K_{22}\right)=840$. Therefore, ${\Gamma }_c\left(G\right)$ is not CN-hyperenergetic.

We have $PSL(2,4)\cong A_5$. Therefore, if $G\cong A_5$ then it follows that ${\Gamma }_c\left(G\right)$ is not CN-hyperenergetic (follows from Theorem 11).

If $G\cong S_4$ then the characteristic polynomial of $CN\left({\Gamma }_c\left(G\right)\right)$ is given by $x^8{(x-3)}^2{(x+1)}^{11}(x^2-5x-30)$ and so

$$CN-spec\left({\Gamma }_c\left(G\right)\right)=\left\{0^8,3^2,{(-1)}^{11},{\left(\frac{5+\sqrt{145}}{2}\right)}^1,{\left(\frac{5-\sqrt{145}}{2}\right)}^1\right\}.$$

Therefore, $E_{cn}\left({\Gamma }_c\left(G\right)\right)=17+\sqrt{145}$. Additionally, by Lemma 1, we have $E_{cn}\left(K_{23}\right)=924$. Therefore, ${\Gamma }_c\left(G\right)$ is not CN-hyperenergetic. Hence, the result follows from [38] (Theorem 2.2). □

Theorem 16.

Let G be a finite non-abelian group. If ${\Gamma }_c\left(G\right)$ is toroidal then ${\Gamma }_c\left(G\right)$ is not CN-hyperenergetic.

Proof. 

If $G\cong D_{14},D_{16}$ or $Q_{16}$ then by Corollary 5 it follows that ${\Gamma }_c\left(G\right)$ is not CN-hyperenergetic. If $G\cong Q{D}_{16}$ then, by Theorem 11, we have that ${\Gamma }_c\left(G\right)$ is not CN-hyperenergetic. If G is isomorphic to ${\mathbb{Z}}_7⋊{\mathbb{Z}}_3$ then ${\Gamma }_c\left(G\right)$ is not CN-hyperenergetic, follows from Theorem 11 considering $p=3$ and $r=7$. If $G\cong D_6\times {\mathbb{Z}}_3$ then $\frac{G}{Z\left(G\right)}\cong D_6$. Therefore, by Theorem 7, ${\Gamma }_c\left(G\right)$ is not CN-hyperenergetic. If $G\cong A_4\times {\mathbb{Z}}_2$ then it can be seen that ${\Gamma }_c\left(G\right)=K_6\bigsqcup 4K_4$. Therefore, by Theorem 2, we have $E_{cn}\left({\Gamma }_c\left(G\right)\right)=2(5·4+4·3·2)=88$. Also, by Lemma 1, we have $E_{cn}\left(K_{22}\right)=2·21·20=840$. Hence, ${\Gamma }_c\left(G\right)$ is not CN-hyperenergetic. Hence, the result follows from [39] (Theorem 6.6). □

We also have the following result.

Theorem 17.

Let G be a finite non-abelian group. If the complement of ${\Gamma }_c\left(G\right)$ is planar then ${\Gamma }_c\left(G\right)$ is not CN-hyperenergetic.

Proof. 

The result follows from [40] (Proposition 2.3) and Corollary 5. □

In view of the above results we conclude this paper with a few conjectures.

Conjecture 1.

A planar or toroidal graph is not CN-hyperenergetic.

Conjecture 2.

${\Gamma }_c\left(G\right)$ is not CN-hyperenergetic.

Conjecture 3.

If $\mathcal{G}=l_1{K}_{m_1}\bigsqcup l_2{K}_{m_2}\bigsqcup \cdots \bigsqcup l_k{K}_{m_k}$, where $l_i{K}_{m_i}$ denotes the disjoint union of $l_i$ copies of the complete graphs $K_{m_i}$ on $m_i$ vertices for $1\le i\le k$, then it is not CN-hyperenergetic.