Laplacian Spectrum of non-commuting graphs of finite groups

TL;DR

This paper computes the Laplacian spectrum of non-commuting graphs for certain finite groups, showing they are L-integral and establishing conditions for this property.

Contribution

It provides explicit spectral computations for non-commuting graphs of specific finite groups and identifies conditions for their L-integrality.

Findings

Non-commuting graphs of studied groups are L-integral

Spectral properties depend on group structure

Conditions for L-integrality of non-commuting graphs

Abstract

In this paper, we compute the Laplacian spectrum of non-commuting graphs of some classes of finite non-abelian groups. Our computations reveal that the non-commuting graphs of all the groups considered in this paper are L-integral. We also obtain some conditions on a group $G$ so that its non-commuting graph is L-integral.

Equations

$$L-Spec-spec(\mathcal{G})=\left\{0^{{\sum }_{i=1}^k{l}_i},m_1^{l_1(m_1-1)},m_2^{l_2(m_2-1)},\dots ,m_k^{l_k(m_k-1)}\right\}.$$

$$\{0,(|V(\mathcal{G})|-{\alpha }_n{)}^{a_n},(|V(\mathcal{G})|-{\alpha }_{n-1}{)}^{a_{n-1}},(|V(\mathcal{G})|-{\alpha }_{n-2}{)}^{a_{n-2}},\dots ,(|V(\mathcal{G})|-{\alpha }_1{)}^{a_1-1}\}.$$

$$L-Spec-spec(\overline{\mathcal{G}})=$$

$$\dots ,{\left(\sum _{i=1}^k{l}_i{m}_i-m_1\right)}^{l_1(m_1-1)},{\left(\sum _{i=1}^k{l}_i{m}_i\right)}^{{\sum }_{i=1}^k{l}_i-1}\}.$$

$$L-Spec-spec({\mathcal{A}}_G)=$$

$$(|G|-|X_1|{)}^{|X_1|-|Z(G)|-1},(|G|-|Z(G)|{)}^{n-1}\}.$$

$$\overline{{\mathcal{A}}_G}=\stackrel{n}{\bigsqcup _{i=1}}{K}_{|X_i|-|Z(G)|}.$$

$$L-Spec-spec({\mathcal{A}}_G)=\{0,{\left(\sum _{i=1}^n(|X_i|-|Z(G)|)-(|X_n|-|Z(G)|)\right)}^{|X_n|-|Z(G)|-1},\dots ,$$

$${\left(\sum _{i=1}^n(|X_i|-|Z(G)|)-(|X_1|-|Z(G)|)\right)}^{|X_1|-|Z(G)|-1},{\left(\sum _{i=1}^n(|X_i|-|Z(G)|)\right)}^{n-1}\}.$$

$$L-Spec-spec({\mathcal{A}}_{G\times A})=$$

$$(|A|(|G|-|X_1|){)}^{|A|(|X_1|-|Z(G)|)-1},(|A|(|G|-|Z(G)|){)}^{n-1}\}.$$

$$L-Spec-spec({\mathcal{A}}_G)=\{0,{(15|Z(G)|)}^{4|Z(G)|-1},{(16|Z(G)|)}^{15|Z(G)|-5},{(19|Z(G)|)}^5\}.$$

$$\frac{G}{Z(G)}=⟨aZ(G),bZ(G):a^{5}Z(G)=b^{4}Z(G)=Z(G),b^{-1}abZ(G)=a^{2}Z(G)⟩.$$

$$\begin{array}[]{ll}C_{G}(ab)&=Z(G)\sqcup abZ(G)\sqcup a^{4}b^{2}Z(G)\sqcup a^{3}b^{3}Z(G),\\ C_{G}(a^{2}b)&=Z(G)\sqcup a^{2}bZ(G)\sqcup a^{3}b^{2}Z(G)\sqcup ab^{3}Z(G),\\ C_{G}(a^{2}b^{3})&=Z(G)\sqcup a^{2}b^{3}Z(G)\sqcup ab^{2}Z(G)\sqcup a^{4}bZ(G),\\ C_{G}(b)&=Z(G)\sqcup bZ(G)\sqcup b^{2}Z(G)\sqcup b^{3}Z(G),\\ C_{G}(a^{3}b)&=Z(G)\sqcup a^{3}bZ(G)\sqcup a^{2}b^{2}Z(G)\sqcup a^{4}b^{3}Z(G)\quad\text{ and }\\ C_{G}(a)&=Z(G)\sqcup aZ(G)\sqcup a^{2}Z(G)\sqcup a^{3}Z(G)\sqcup a^{4}Z(G)\\ \end{array}$$

$$|C_G(ab)|=|C_G(a^{2}b)|=|C_G(a^2{b}^3)|=|C_G(b)|=|C_G(a^{3}b)|=4|Z(G)|.$$

$$L-Spec-spec({\mathcal{A}}_G)=\{0,{((p^2-p)|Z(G)|)}^{(p^2-1)|Z(G)|-p-1},{((p^2-1)|Z(G)|)}^p\}.$$

$$C_G(a)$$

$$C_G(a^{j}b)$$

$$L-Spec-spec({\mathcal{A}}_G)=\{0,{(p^3-p^2)}^{p^3-2p-1},{(p^3-p)}^p\}.$$

$$L-Spec-spec({\mathcal{A}}_G)=$$

$${((2m-1)|Z(G)|)}^m\}.$$

$$C_G(x{y}^j)=C_G(x{y}^{j}z)=Z(G)\bigsqcup x{y}^{j}Z(G),1\le j\le m$$

$$C_G(y)=C_G(y^{i}z)=Z(G)\bigsqcup yZ(G)\bigsqcup \cdots \bigsqcup y^{m-1}Z(G),1\le i\le m-1$$

$$=\{\begin{array}{ll}\{0,(mn{)}^{mn-n-1},{(2mn-2n)}^{mn-m},{(2mn-n)}^m\}& \text{if m is odd}\\ \{0,(mn{)}^{mn-2n-1},{(2mn-4n)}^{mn-\frac{m}{2}},{(2mn-2n)}^{\frac{m}{2}}\}& \text{if m is even}.\end{array}$$

$$L-Spec-spec({\mathcal{A}}_{D_{2m}})=\{\begin{array}{ll}\{0,m^{m-2},{(2m-1)}^m\}& \text{if m is odd}\\ \{0,m^{m-3},{(2m-4)}^{\frac{m}{2}},(2m-2{)}^{\frac{m}{2}}\}& \text{if m is even}.\end{array}$$

$$L-Spec-spec({\mathcal{A}}_{Q_{4n}})=\{0,{(2n)}^{2n-3},{(4n-4)}^n,{(4n-2)}^n\}.$$

$$L-Spec-spec({\mathcal{A}}_G)=\{0,{(pq-q)}^{q-2},{(pq-p)}^{pq-2q},(pq-1{)}^q\}.$$

$$L-Spec-spec({\mathcal{A}}_{Q{D}_{2^n}})=\{0,{(2^{n-1})}^{2^{n-1}-3},{(2^n-4)}^{2^{n-2}},{(2^n-2)}^{2^{n-2}}\}.$$

$$C_{Q{D}_{2^n}}(a)=C_{Q{D}_{2^n}}(a^i)=⟨a⟩\text{ for }1\le i\le 2^{n-1}-1,i\ne 2^{n-2}$$

$$C_{Q{D}_{2^n}}(a^{j}b)=\{1,a^{2^{n-2}},a^{j}b,a^{j+2^{n-2}}b\}\text{ for }1\le j\le 2^{n-2}$$

$$L-Spec-spec({\mathcal{A}}_{PSL(2,2^k)})=$$

$${(2^{3k}-2^{k+1}+1)}^{2^{3k-1}-2^{3k}-3.2^{k-1}},{(2^{3k}-2^k-1)}^{2^{2k}+2^k}\}.$$

$$\{xP{x}^{-1},xA{x}^{-1},xB{x}^{-1}:x\in PSL(2,2^k)\}$$

$$\overline{{\mathcal{A}}_{PSL(2,2^k)}}=(2^k+1)K_{|xP{x}^{-1}|-1}\bigsqcup 2^{k-1}(2^k+1)K_{|xA{x}^{-1}|-1}\bigsqcup 2^{k-1}(2^k-1)K_{|xB{x}^{-1}|-1}.$$

Full text

Laplacian Spectrum of non-commuting graphs of finite groups

Parama Dutta, Jutirekha Dutta and Rajat Kanti Nath111Corresponding author

Department of Mathematical Sciences,

Tezpur University, Napaam-784028, Sonitpur, Assam, India.

Emails: parama@gonitsora.com, jutirekhadutta@yahoo.com and rajatkantinath@yahoo.com

**Abstract: In this paper, we compute the Laplacian spectrum of non-commuting graphs of some classes of finite non-abelian groups. Our computations reveal that the non-commuting graphs of all the groups considered in this paper are L-integral. We also obtain some conditions on a group $G$ so that its non-commuting graph is L-integral. **

Key words: non-commuting graph, spectrum, L-integral graph, finite group.

2010 Mathematics Subject Classification: 05C50, 15A18, 05C25, 20D60.

1 Introduction

Let $G$ be a finite group with centre $Z(G)$. The non-commuting graph of a non-abelian group $G$, denoted by ${\mathcal{A}}_{G}$, is a simple undirected graph whose vertex set is $G\setminus Z(G)$ and two vertices $x$ and $y$ are adjacent if and only if $xy\neq yx$. Various aspects of non-commuting graphs of different finite groups can be found in [1, 4, 8, 12, 23]. In [12], Elvierayani and Abdussakir have computed the Laplacian spectrum of the non-commuting graph of dihedral groups $D_{2m}$ where $m$ is odd and suggested to consider the case when $m$ is even. In this paper, we compute the Laplacian spectrum of the non-commuting graph of $D_{2m}$ for any $m\geq 3$ using a different method. Our method also enables to compute the Laplacian spectrum of the non-commuting graphs of several well-known families finite non-abelian groups such as the quasidihedral groups, generalized quaternion groups, some projective special linear groups, general linear groups etc. In a separate paper [11], we study the Laplacian energy of non-commuting graphs of the groups considered in this paper.

For a graph ${\mathcal{G}}$ we write $\overline{\mathcal{G}}$ and $V({\mathcal{G}})$ to denote the complement of ${\mathcal{G}}$ and the set of vertices of ${\mathcal{G}}$ respectively. Let $A({\mathcal{G}})$ and $D({\mathcal{G}})$ denote the adjacency matrix and degree matrix of a graph ${\mathcal{G}}$ respectively. Then the Laplacian matrix of ${\mathcal{G}}$ is given by $L({\mathcal{G}})=D({\mathcal{G}})-A({\mathcal{G}})$. We write $\operatorname{L-Spec}-spec({\mathcal{G}})$ to denote the Laplacian spectrum of ${\mathcal{G}}$ and $\operatorname{L-Spec}-spec({\mathcal{G}})=\{\alpha_{1}^{a_{1}},\alpha_{2}^{a_{2}},\dots,\alpha_{n}^{a_{n}}\}$ where $\alpha_{1}<\alpha_{2}<\cdots<\alpha_{n}$ are the eigenvalues of $L({\mathcal{G}})$ with multiplicities $a_{1},a_{2},\dots,a_{n}$ respectively. A graph ${\mathcal{G}}$ is called L-integral if $\operatorname{L-Spec}-spec({\mathcal{G}})$ contains only integers. As a consequence of our results, it follows that the non-commuting graphs of all the groups considered in this paper are L-integral. It is worth mentioning that L-integral graphs are studied extensively in [3, 15, 17].

2 Preliminary results

It is well-known that $\operatorname{L-Spec}-spec(K_{n})=\{0^{1},n^{n-1}\}$ where $K_{n}$ denotes the complete graph on $n$ vertices. Further, we have the following results.

Theorem 2.1**.**

If $\mathcal{G}=l_{1}K_{m_{1}}\sqcup l_{2}K_{m_{2}}\sqcup\cdots\sqcup l_{k}K_{m_{k}}$, where $l_{i}K_{m_{i}}$ denotes the disjoint union of $l_{i}$ copies of $K_{m_{i}}$ for $1\leq i\leq k$ and $m_{1}<m_{2}<\cdots<m_{k}$, then

[TABLE]

Theorem 2.2**.**

[18, Theorem 3.6]* Let $\mathcal{G}$ be a graph such that $\operatorname{L-Spec}-spec({\mathcal{G}})=\{\alpha_{1}^{a_{1}},\alpha_{2}^{a_{2}}$, …, $\alpha_{n}^{a_{n}}\}$ then $\operatorname{L-Spec}-spec(\overline{\mathcal{G}})$ is given by*

[TABLE]

As a corollary of the above two theorems we have the following result.

Corollary 2.3**.**

If $\mathcal{G}=l_{1}K_{m_{1}}\sqcup l_{2}K_{m_{2}}\sqcup\cdots\sqcup l_{k}K_{m_{k}}$, where $l_{i}K_{m_{i}}$ denotes the disjoint union of $l_{i}$ copies of $K_{m_{i}}$ for $1\leq i\leq k$ and $m_{1}<m_{2}<\cdots<m_{k}$, then

[TABLE]

A group $G$ is called an AC-group if $C_{G}(x)$ is abelian for all $x\in G\setminus Z(G)$. Various aspects of AC-groups can be found in [1, 10, 21]. The following result gives the Laplacian spectrum of the non-commuting graph of a finite non-abelian AC-group.

Theorem 2.4**.**

Let $G$ be a finite non-abelian AC-group. Then

[TABLE]

where $X_{1},\dots,X_{n}$ are the distinct centralizers of non-central elements of $G$ such that $|X_{1}|\leq\cdots\leq|X_{n}|$.

Proof.

Let $G$ be a finite non-abelian AC-group and $X_{i}=C_{G}(x_{i})$ where $x_{i}\in G\setminus Z(G)$ and $1\leq i\leq n$. Let $x,y\in X_{i}\setminus Z(G)$ for some $i$ and $x\neq y$ then, since $G$ an AC-group, there is an edge between $x$ and $y$ in $\overline{{\mathcal{A}}_{G}}$. Suppose that $x\in(X_{i}\cap X_{j})\setminus Z(G)$ for some $1\leq i\neq j\leq n$. Then $[x,x_{i}]=1$ and $[x,x_{j}]=1$. Let $s\in C_{G}(x)$ then $[s,x_{i}]=1$ since $x_{i}\in C_{G}(x)$ and $G$ is an AC-group. Therefore, $s\in C_{G}(x_{i})$ and so $C_{G}(x)\subseteq C_{G}(x_{i})$. Again, let $t\in C_{G}(x_{i})$ then $[t,x]=1$ since $x\in C_{G}(x_{i})$ and $G$ is an AC-group. Therefore, $t\in C_{G}(x)$ and so $C_{G}(x_{i})\subseteq C_{G}(x)$. Thus $C_{G}(x)=C_{G}(x_{i})$. Similarly, it can be seen that $C_{G}(x)=C_{G}(x_{j})$, which is a contradiction. Therefore, $X_{i}\cap X_{j}=Z(G)$ for any $1\leq i\neq j\leq n$. This shows that

[TABLE]

Therefore, by Corollary 2.3, we have

[TABLE]

Hence, the result follows noting that $\underset{i=1}{\overset{n}{\sum}}(|X_{i}|-|Z(G)|)=|G|-|Z(G)|$. ∎

Corollary 2.5**.**

*Let $G$ be a finite non-abelian AC-group and $A$ be any finite abelian group. Then *

[TABLE]

where $X_{1},\dots,X_{n}$ are the distinct centralizers of non-central elements of $G$ such that $|X_{1}|\leq\cdots\leq|X_{n}|$.

Proof.

It is easy to see that $G\times A$ is an AC-group and $X_{1}\times A,X_{2}\times A,\dots,X_{n}\times A$ are the distinct centralizers of non-central elements of $G\times A$. Hence, the result follows from Theorem 2.4 noting that $Z(G\times A)=Z(G)\times A$. ∎

3 Groups with given central factors

In this section, we compute the Laplacian spectrum of the non-commuting graphs of some families of finite non-abelian groups whose central factors are some well-known finite groups. We begin with the following.

Theorem 3.1**.**

Let $G$ be a finite group and $\frac{G}{Z(G)}\cong Sz(2)$, where $Sz(2)$ is the Suzuki group presented by $\langle a,b:a^{5}=b^{4}=1,b^{-1}ab=a^{2}\rangle$. Then

[TABLE]

Proof.

We have

[TABLE]

Observe that

[TABLE]

are the only centralizers of non-central elements of $G$. Also note that these centralizers are abelian subgroups of $G$. Thus $G$ is an AC-group. We have $|C_{G}(a)|=5|Z(G)|$ and

[TABLE]

Therefore, by Theorem 2.4, the result follows. ∎

Theorem 3.2**.**

Let $G$ be a finite group such that $\frac{G}{Z(G)}\cong{\mathbb{Z}}_{p}\times{\mathbb{Z}}_{p}$, where $p$ is a prime integer. Then

[TABLE]

Proof.

Let $|Z(G)|=n$ then since $\frac{G}{Z(G)}\cong{\mathbb{Z}}_{p}\times{\mathbb{Z}}_{p}$ we have $\frac{G}{Z(G)}=\langle aZ(G),bZ(G):a^{p},b^{p},aba^{-1}b^{-1}\in Z(G)\rangle$, where $a,b\in G$ with $ab\neq ba$. Then for any $z\in Z(G)$, we have

[TABLE]

These are the only centralizers of non-central elements of $G$. Also note that these centralizers are abelian subgroups of $G$. Therefore, $G$ is an AC-group. We have $|C_{G}(a)|=|C_{G}(a^{j}b)|=pn$ for $1\leq j\leq p$. Hence, the result follows from Theorem 2.4. ∎

As a corollary we have the following result.

Corollary 3.3**.**

Let $G$ be a non-abelian group of order $p^{3}$, for any prime $p$, then

[TABLE]

Proof.

Note that $|Z(G)|=p$ and $\frac{G}{Z(G)}\cong{\mathbb{Z}}_{p}\times{\mathbb{Z}}_{p}$. Hence the result follows from Theorem 3.2. ∎

Theorem 3.4**.**

Let $G$ be a finite group such that $\frac{G}{Z(G)}\cong D_{2m}$, for $m\geq 2$. Then

[TABLE]

Proof.

Since $\frac{G}{Z(G)}\cong D_{2m}$ we have $\frac{G}{Z(G)}=\langle xZ(G),yZ(G):x^{2},y^{m},xyx^{-1}y\in Z(G)\rangle$, where $x,y\in G$ with $xy\neq yx$. It is not difficult to see that for any $z\in Z(G)$,

[TABLE]

and

[TABLE]

are the only centralizers of non-central elements of $G$. Also note that these centralizers are abelian subgroups of $G$. Therefore, $G$ is an AC-group. We have $|C_{G}(x^{j}y)|=2n$ for $1\leq j\leq m$ and $|C_{G}(y)|=mn$, where $|Z(G)|=n$. Hence, the result follows from Theorem 2.4. ∎

Using Theorem 3.4, we now compute the Laplacian spectrum of the non-commuting graphs of the groups $M_{2mn},D_{2m}$ and $Q_{4n}$ respectively.

Corollary 3.5**.**

Let $M_{2mn}=\langle a,b:a^{m}=b^{2n}=1,bab^{-1}=a^{-1}\rangle$ be a metacyclic group, where $m>2$. Then $\operatorname{L-Spec}-spec({\mathcal{A}}_{M_{2mn}})$

[TABLE]

Proof.

Observe that $Z(M_{2mn})=\langle b^{2}\rangle$ or $\langle b^{2}\rangle\cup a^{\frac{m}{2}}\langle b^{2}\rangle$ according as $m$ is odd or even. Also, it is easy to see that $\frac{M_{2mn}}{Z(M_{2mn})}\cong D_{2m}$ or $D_{m}$ according as $m$ is odd or even. Hence, the result follows from Theorem 3.4. ∎

As a corollary to the above result we have the following result.

Corollary 3.6**.**

Let $D_{2m}=\langle a,b:a^{m}=b^{2}=1,bab^{-1}=a^{-1}\rangle$ be the dihedral group of order $2m$, where $m>2$. Then

[TABLE]

Corollary 3.7**.**

Let $Q_{4n}=\langle x,y:y^{2n}=1,x^{2}=y^{n},xyx^{-1}=y^{-1}\rangle$, where $n\geq 2$, be the generalized quaternion group of order $4n$. Then

[TABLE]

Proof.

The result follows from Theorem 3.4 noting that $Z(Q_{4n})=\{1,a^{n}\}$ and $\frac{Q_{4n}}{Z(Q_{4n})}$ $\cong D_{2n}$. ∎

4 Some well-known groups

In this section, we compute the Laplacian spectrum of the non-commuting graphs of some well-known families of finite groups. We begin with the family of finite groups having order $pq$ where $p$ and $q$ are primes.

Proposition 4.1**.**

Let $G$ be a non-abelian group of order $pq$, where $p$ and $q$ are primes with $p\mid(q-1)$. Then

[TABLE]

Proof.

It is easy to see that $|Z(G)|=1$ and $G$ is an AC-group. Also the centralizers of non-central elements of $G$ are precisely the Sylow subgroups of $G$. The number of Sylow $q$-subgroups and Sylow $p$-subgroups of $G$ are one and $q$ respectively. Hence, the result follows from Theorem 2.4. ∎

Proposition 4.2**.**

The Laplacian spectrum of the non-commuting graph of the quasidihedral group $QD_{2^{n}}=\langle a,b:a^{2^{n-1}}=b^{2}=1,bab^{-1}=a^{2^{n-2}-1}\rangle$, where $n\geq 4$, is given by

[TABLE]

Proof.

It is well-known that $Z(QD_{2^{n}})=\{1,a^{2^{n-2}}\}$. Also

[TABLE]

and

[TABLE]

are the only centralizers of non-central elements of $QD_{2^{n}}$. Note that these centralizers are abelian subgroups of $QD_{2^{n}}$. Therefore, $QD_{2^{n}}$ is an AC-group. We have $|C_{QD_{2^{n}}}(a)|=2^{n-1}$ and $|C_{QD_{2^{n}}}(a^{j}b)|=4$ for $1\leq j\leq 2^{n-2}$. Hence, the result follows from Theorem 2.4. ∎

Proposition 4.3**.**

The Laplacian spectrum of the non-commuting graph of the projective special linear group $PSL(2,2^{k})$, where $k\geq 2$, is given by

[TABLE]

Proof.

We know that $PSL(2,2^{k})$ is a non-abelian group of order $2^{k}(2^{2k}-1)$ with trivial center. By Proposition 3.21 of [1], the set of centralizers of non-trivial elements of $PSL(2,2^{k})$ is given by

[TABLE]

where $P$ is an elementary abelian  $2$-subgroup and $A,\quad B$ are cyclic subgroups of $PSL(2,2^{k})$ having order $2^{k},2^{k}-1$ and $2^{k}+1$ respectively. Also the number of conjugates of $P,A$ and $B$ in $PSL(2,2^{k})$ are $2^{k}+1,2^{k-1}(2^{k}+1)$ and $2^{k-1}(2^{k}-1)$ respectively. Note that $PSL(2,2^{k})$ is a AC-group and so, by (2.1), we have

[TABLE]

That is, $\overline{{\mathcal{A}}_{PSL(2,2^{k})}}=(2^{k}+1)K_{2^{k}-1}\sqcup 2^{k-1}(2^{k}+1)K_{2^{k}-2}\sqcup 2^{k-1}(2^{k}-1)K_{2^{k}}$. Hence, the result follows from Corollary 2.3. ∎

Proposition 4.4**.**

The Laplacian spectrum of the non-commuting graph of the general linear group $GL(2,q)$, where $q=p^{n}>2$ and $p$ is a prime integer, is given by

[TABLE]

Proof.

We have $|GL(2,q)|=(q^{2}-1)(q^{2}-q)$ and $|Z(GL(2,q))|=q-1$. By Proposition 3.26 of [1], the set of centralizers of non-central elements of $GL(2,q)$ is given by

[TABLE]

where $D$ is the subgroup of $GL(2,q)$ consisting of all diagonal matrices, $I$ is a cyclic subgroup of $GL(2,q)$ having order $q^{2}-1$ and $P$ is the Sylow $p$-subgroup of $GL(2,q)$ consisting of all upper triangular matrices with $1$ in the diagonal. The orders of $D$ and $PZ(GL(2,q))$ are $(q-1)^{2}$ and $q(q-1)$ respectively. Also the number of conjugates of $D,I$ and $PZ(GL(2,q))$ in $GL(2,q)$ are $\frac{q(q+1)}{2},\frac{q(q-1)}{2}$ and $q+1$ respectively. Since $GL(2,q)$ is an AC-group (see Lemma 3.5 of [1]), by (2.1), we have $\overline{{\mathcal{A}}_{GL(2,q)}}=$

[TABLE]

That is, $\overline{{\mathcal{A}}_{GL(2,q)}}=\frac{q(q+1)}{2}K_{q^{2}-3q+2}\sqcup\frac{q(q-1)}{2}K_{q^{2}-q}\sqcup(q+1)K_{q^{2}-2q+1}$. Hence, the result follows from Corollary 2.3. ∎

Proposition 4.5**.**

Let $F=GF(2^{n}),n\geq 2$ and $\vartheta$ be the Frobenius automorphism of $F$, that is, $\vartheta(x)=x^{2}$ for all $x\in F$. Then the Laplacian spectrum of the non-commuting graph of the group

[TABLE]

under matrix multiplication given by $U(a,b)U(a^{\prime},b^{\prime})=U(a+a^{\prime},b+b^{\prime}+a^{\prime}\vartheta(a))$ is

[TABLE]

Proof.

Note that $Z(A(n,\vartheta))=\{U(0,b):b\in F\}$ and so $|Z(A(n,\vartheta))|=2^{n}$. Let $U(a,b)$ be a non-central element of $A(n,\vartheta)$. It can be seen that the centralizer of $U(a,b)$ in $A(n,\vartheta)$ is $Z(A(n,\vartheta))\sqcup U(a,0)Z(A(n,\vartheta))$. Clearly $A(n,\vartheta)$ is an AC-group and so, by (2.1), we have $\overline{{\mathcal{A}}_{A(n,\vartheta)}}=(2^{n}-1)K_{2^{n}}$. Hence the result follows from Corollary 2.3. ∎

Proposition 4.6**.**

Let $F=GF(p^{n})$, $p$ be a prime. Then the Laplacian spectrum of the non-commuting graph of the group

[TABLE]

under matrix multiplication $V(a,b,c)V(a^{\prime},b^{\prime},c^{\prime})=V(a+a^{\prime},b+b^{\prime}+ca^{\prime},c+c^{\prime})$ is

[TABLE]

Proof.

We have $Z(A(n,p))=\{V(0,b,0):b\in F\}$ and so $|Z(A(n,p))|=p^{n}$. The centralizers of non-central elements of $A(n,p)$ are given by

1. (i)

If $b,c\in F$ and $c\neq 0$ then the centralizer of $V(0,b,c)$ in $A(n,p)$ is

$\{V(0,b^{\prime},c^{\prime}):b^{\prime},c^{\prime}\in F\}$ having order $p^{2n}$. 2. (ii)

If $a,b\in F$ and $a\neq 0$ then the centralizer of $V(a,b,0)$ in $A(n,p)$ is

$\{V(a^{\prime},b^{\prime},0):a^{\prime},b^{\prime}\in F\}$ having order $p^{2n}$. 3. (iii)

If $a,b,c\in F$ and $a\neq 0,c\neq 0$ then the centralizer of $V(a,b,c)$ in $A(n,p)$ is $\{V(a^{\prime},b^{\prime},ca^{\prime}a^{-1}):a^{\prime},b^{\prime}\in F\}$ having order $p^{2n}$.

It can be seen that all the centralizers of non-central elements of $A(n,p)$ are abelian. Hence $A(n,p)$ is an AC-group and so, by (2.1), we have

[TABLE]

Hence the result follows from Corollary 2.3. ∎

We would like to mention here that the groups considered in Proposition 4.5-4.6 are constructed by Hanaki (see [14]). These groups are also considered in [5], in order to compute their numbers of distinct centralizers.

5 Some consequences

Note that the non-commuting graphs of all the groups considered in Section 3 and 4 are L-integral. In this section, we determine some conditions on $G$ so that its non-commuting graph becomes L-integral.

A finite group is called an $n$-centralizer group if it has $n$ numbers of distinct element centralizers. It clear that $1$-centralizer groups are precisely the abelian groups. There are no $2$, $3$-centralizer finite groups. The study of these groups was initiated by Belcastro and Sherman [6] in the year 1994. We have the following results regarding $n$-centralizer groups.

Proposition 5.1**.**

If $G$ is a finite $4$-centralizer group then ${\mathcal{A}}_{G}$ is L-integral.

Proof.

Let $G$ be a finite $4$-centralizer group. Then, by [6, Theorem 2], we have $\frac{G}{Z(G)}\cong{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$. Therefore, by Theorem 3.2, we have

[TABLE]

Hence, ${\mathcal{A}}_{G}$ is L-integral. ∎

Further, we have the following result.

Proposition 5.2**.**

If $G$ is a finite $(p+2)$-centralizer $p$-group for any prime $p$, then ${\mathcal{A}}_{G}$ is L-integral.

Proof.

Let $G$ be a finite $(p+2)$-centralizer $p$-group. Then, by [5, Lemma 2.7], we have $\frac{G}{Z(G)}\cong{\mathbb{Z}}_{p}\times{\mathbb{Z}}_{p}$. Therefore, by Theorem 3.2, we have

[TABLE]

Hence, ${\mathcal{A}}_{G}$ is L-integral. ∎

Proposition 5.3**.**

If $G$ is a finite $5$-centralizer group then ${\mathcal{A}}_{G}$ is L-integral.

Proof.

Let $G$ be a finite $5$-centralizer group. Then by [6, Theorem 4] we have $\frac{G}{Z(G)}\cong{\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}$ or $D_{6}$. Now, if $\frac{G}{Z(G)}\cong{\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}$ then by Theorem 3.2 we have $\operatorname{L-Spec}-spec({\mathcal{A}}_{G})=\{0,{(6|Z(G)|)}^{8|Z(G)|-4},{(8|Z(G)|)}^{3}\}$ and hence ${\mathcal{A}}_{G}$ is L-integral. If $\frac{G}{Z(G)}\cong D_{6}$ then, by Theorem 3.4, we have

[TABLE]

and hence ${\mathcal{A}}_{G}$ is L-integral. Therefore, the result follows. ∎

We also have the following corollary.

Corollary 5.4**.**

Let $G$ be a finite non-abelian group and $\{x_{1},x_{2},\dots,x_{r}\}$ be a set of pairwise non-commuting elements of $G$ having maximal size. Then ${\mathcal{A}}_{G}$ is L-integral if $r=3,4$.

Proof.

By Lemma 2.4 in [2], we have that $G$ is a $4$-centralizer or a $5$-centralizer group according as $r=3$ or $4$. Hence the result follows from Proposition 5.1 and Proposition 5.3. ∎

The commuting probability of a finite group $G$ denoted by $\Pr(G)$ is the probability that any two randomly chosen elements of $G$ commute. Clearly, $\Pr(G)=1$ if and only if $G$ is abelian. The study of $\Pr(G)$ is originated from a paper of Erd$\ddot{\rm o}$s and Tur$\acute{\rm a}$n [13]. Various results on $\Pr(G)$ can be found in [7, 9, 19]. The following results show that ${\mathcal{A}}_{G}$ is L-integral if $\Pr(G)$ has some particular values.

Proposition 5.5**.**

If $\Pr(G)\in\{\frac{5}{14},\frac{2}{5},\frac{11}{27},\frac{1}{2},\frac{5}{8}\}$ then ${\mathcal{A}}_{G}$ is L-integral.

Proof.

If $\Pr(G)\in\{\frac{5}{14},\frac{2}{5},\frac{11}{27},\frac{1}{2},\frac{5}{8}\}$ then as shown in [22, pp. 246] and [20, pp. 451], we have $\frac{G}{Z(G)}$ is isomorphic to one of the groups in $\{D_{14},D_{10},D_{8},D_{6},{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}\}$. If $\frac{G}{Z(G)}$ is isomorphic to $D_{14},D_{10},D_{8}$ or $D_{6}$ then, by Theorem 3.4, it follows that ${\mathcal{A}}_{G}$ is L-integral. If $\frac{G}{Z(G)}$ is isomorphic to ${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ then, by Theorem 3.2, it follows that ${\mathcal{A}}_{G}$ is L-integral. Hence, the result follows. ∎

Proposition 5.6**.**

Let $G$ be a finite group and $p$ the smallest prime divisor of $|G|$. If $\Pr(G)=\frac{p^{2}+p-1}{p^{3}}$ then ${\mathcal{A}}_{G}$ is L-integral.

Proof.

If $\Pr(G)=\frac{p^{2}+p-1}{p^{3}}$ then by [16, Theorem 3] we have $\frac{G}{Z(G)}$ is isomorphic to ${\mathbb{Z}}_{p}\times{\mathbb{Z}}_{p}$. Now, by Theorem 3.2, it follows that ${\mathcal{A}}_{G}$ is L-integral. ∎

Proposition 5.7**.**

If $G$ is a non-solvable group with $\Pr(G)=\frac{1}{12}$ then ${\mathcal{A}}_{G}$ is L-integral.

Proof.

By [7, Proposition 3.3.7], we have that $G$ is isomorphic to $A_{5}\times B$ for some abelian group $B$. Since $A_{5}$ is an AC-group, by Corollary 2.5, it follows that ${\mathcal{A}}_{G}$ is L-integral. ∎

A graph is called planar if it can be embedded in the plane so that no two edges intersect geometrically except at a vertex to which both are adjacent. We conclude this paper with the following result.

Proposition 5.8**.**

Let $G$ be a finite group then ${\mathcal{A}}_{G}$ is L-integral if ${\mathcal{A}}_{G}$ is planar.

Proof.

It was shown in Proposition 2.3 of [1] that ${\mathcal{A}}_{G}$ is planar if and only if $G$ is isomorphic to $D_{6},D_{8}$ or $Q_{8}$. Therefore, by Corollary 3.6 and Corollary 3.7, the result follows. ∎