$A_{\alpha}$ matrix of commuting graphs of non-abelian groups

1.   Introduction

In this article, all our graphs are simple, finite and connected. A graph $G = G(V(G), E(G))$ is an ordered pair consisting of the vertex set $V(G)$ and the edge set $E(G)$. The cardinality of $V(G)$ is referred to as the order of $G$ represented by $n$ and the cardinality of $E(G)$ is called the size of $G$, symbolized by $m$. The set of vertices adjacent to $v \in V(G)$, indicated by $N(v)$, is known as the neighborhood of $v.$ The cardinality of $N(v)$ is the degree of a vertex $v,$ symbolized by $d(v)$. A graph $G$ is called regular if the degree of every vertex is same. For other notations and terminology, see ^[14].

Let $M_{n}(\mathbb{R})$ be the set of square matrices over the field $\mathbb{R}$ and $M^{*}: = \{M\in M_{n}(\mathbb{R}): M^{T} = M\}$, where $M^{*}$ is the set of real symmetric matrices. The adjacency matrix of $G$, symbolized by $A(G)$, is described in the following way:

Clearly, $A(G)$ is in $M^{*},$ so all its eigenvalues are in $\mathbb{R}$ and can be indexed from the largest to the least as: $\lambda_{1}\geq\lambda_{2}\geq \dots\geq \lambda_{n}.$ The multiset of eigenvalues of $A(G)$ is the spectrum of $G$, and $\lambda_{1}$ is called the spectral radius (or spectral norm) of $G$.

Let $D(G) = diag(d(v_{1}), d(v_{2}), \dots, d(v_{n}))$ be the diagonal matrix of vertex degrees of $G.$ The matrices $L(G) = A(G)-D(G)$ and $Q(G) = A(G)+D(G)$ are known as the Laplacian matrix and the signless Laplacian matrix of $G$, respectively. The matrix $L(G)$ is positive semi-definite and the matrix $Q(G)$ is positive semi-definite (definite) and their spectrum is in $\mathbb{R}$. Laplacian eigenvalues are denoted by $\lambda_{1}^{L}\geq\lambda_{2}^{L}\geq \dots\geq \lambda_{n}^{L} = 0$ while the signless Laplacian eigenvalues are denoted by $\lambda_{1}^{Q}\geq\lambda_{2}^{Q}\geq \dots\geq \lambda_{n}^{Q}$. It is well known that $\lambda_{n}^{L} = 0$ is always the eigenvalue of $L(G)$ and $\lambda_{n-1}^{L} > 0$ for connected graphs, which is referred to as the algebraic connectivity of $G.$

Nikiforov ^[23] suggested examining the convex combinations $A_\alpha(G)$ of the adjacency matrix $A(G)$ and the diagonal matrix $D(G)$ specified by $A_{\alpha}(G) = (1-\alpha)A(G) + \alpha D(G)$, where $0\leq \alpha \leq 1$. Obviously, $A_{0}(G) = A(G)$, $A_{1}(G) = D(G)$ and $Q(G) = 2A_{\frac{1}{2}}(G) = A(G) + D(G)$. Also $A_{\alpha}(G)-A_{\gamma}(G) = (\alpha-\gamma)L(G) = (\alpha-\gamma)(D(G)-A(G))$, where $L(G)$ is the Laplacian matrix of $G$. Thus, $A_{\alpha}(G)$ matrix merges the spectral theories of $A(G), \; L(G)$ and $Q(G)$ as well as their uncountably many combinations. Recent articles on the spectral properties of the A matrix can be obtained in ^{[21,23,24,25,27,33]} and the references in those articles.

The matrix $A_{\alpha}(G)$ belongs to the class $M^{*}$, so its eigenvalues can be arranged in decreasing order as $\lambda_{1}^{\alpha}\geq \lambda_{2}^{\alpha}\geq \cdots \geq \lambda_{n}^{\alpha},$ where $\lambda_{1}^{\alpha}$ is called the $\alpha$ spectral radius (or $\lambda (A_{\alpha})$ generalized adjacency spectral radius) of $G$. For a connected graph $G$, the matrix $A_{\alpha}(G)$ (for $\alpha\ne 1$) is an irreducible and non-negative. Consequently, according to the Perron-Frobenius theorem, $\lambda(A_{\alpha}(G))$ is the simple eigenvalue, and $\lambda(A_{\alpha}(G))$ has only one positive unit eigenvector $X$, which is known as the generalized adjacency Perron vector of $G.$

Consider $\mathcal{G}$ is a finite group with $n$ elements and $e$ is the identity element. If $X$ is a non empty subset of $\mathcal{G}$, then the commuting graph of $\mathcal{G}$ related to $X$, is indicated by $\mathcal{C}(\mathcal{G}, X)$, and is defined with $X$ as the vertex set and two different vertices $x$ and $y$ are edge connected in $\mathcal{C}(\mathcal{G}, X)$ if and only if they commute in $X.$ The commuting graphs of matrix rings and semirings over the finite fields were studied in ^[1,15]. The metric dimension, the resolving polynomial, the clique number and the chromatic number of the commuting graphs of the dihedral groups were discussed in ^[4,11]. Recent results on the commuting graphs of the generalized dihedral groups can be found in ^[12,20]. The adjacency spectrum of commuting graphs were studied in ^[5,13], the Laplacian as well as the signless Laplacian spectra of the commuting graphs on the dihedral groups were explored in ^[3,32]. For other spectral properties and energies of commuting graphs, see ^[16,18].

2.   $A_{\alpha}$ eigenvalues of commuting graphs of finite non-abelian groups

Any column vector $X = (x_{1}, x_{2}, \dots, x_{n})^{T} \in \mathbb{R}^{n}$ can be regarded as a function defined by $V(G)$ which associates every vertex $v_{i}$ to $x_{i}$, i.e., $X(v_{i}) = x_{i}$ for all $i = 1, \dots, n.$ Also, the quadratic form of the $A_{\alpha}$ matrix is:

and $\lambda$ is the $A_{\alpha}$ eigenvalue of $G$ that corresponds to the eigenvector $X$ whenever $X\neq 0$ and for every $v_{i}\in V(G)$, we have:

or equivalently

Equations (2.1) and (2.2) are known as eigenequations for the matrix $A_{\alpha}$ of $G.$

Our first result is helpful in finding some $A_{\alpha}$ eigenvalues of $G$ with some special structure.

Theorem 2.1. Suppose $G$ is a graph with $V(G) = \{v_{1}, v_{2}, \dots, v_{n}\}$ and $B = \{v_{1}, v_{2}, \dots, v_{k}\}$ is the set of vertices of $G$ satisfying $N(v_{i}) = N(v_{j})$ for all $i, j\in \{1, 2, \dots, k\}$. Then the following hold.

(ⅰ) If $B$ is an independent set of $G$, then $b\alpha$ is an $A_{\alpha}$ eigenvalue of $G$ with multiplicity at least $k-1,$ where $b$ is the degree of $v_{i}$, for $i = 1, 2, \dots, k.$

(ⅱ) If $B$ is a clique of $G$, then $\alpha(\omega+\beta)-1$ is an $A_{\alpha}$ eigenvalue of $G$ having multiplicity at least $k-1,$ whereas $\beta$ is the total number of vertices in $V(G)\setminus B$, that are edge connected to every vertex of clique.

Proof. (ⅰ) Since $\{v_{1}, v_{2}, \dots, v_{k}\}$ is the independent set of $G$ sharing the same neighbourhood, so $d(v_{1}) = d(v_{2}) = \dots = d(v_{k}) = b$ (say). We first index the independent vertices, so that the $A_{\alpha}$ matrix of $G$ can be put as:

Let $X_{i-1} = \Big(-1, x_{i2}, x_{i3}, \dots, x_{ik}, \underbrace{ 0, 0, 0, \dots, 0}_{n-k} \Big)^{T}$ be the vector in $\mathbb{R}^{n}$ such that

Clearly, $X_{1}, X_{2}, \dots, X_{k-1}$ are the linearly independent vectors and we note that all the rows of $B_{k\times (n-k)}$ are identical. Therefore, we have

Thus, it follows that $X_{1}$ is the eigenvector that corresponds to the $A_{\alpha}$ eigenvalue $b \alpha.$ Similarly, $X_{2}, X_{3}, \dots, X_{p-1}$ are the eigenvectors of the $A_{\alpha}$ matrix corresponding to the eigenvalue $b\alpha.$

(ⅱ) By hypothesis, $\{v_{1}, v_{2}, \dots, v_{k}\}$ forms the clique of $G$ with the same neighbourhood, so $d(v_{1}) = d(v_{2}) = \dots = d(v_{k}) = \omega-1+\beta$, whereas $\beta$ is the number of vertices in $|N(v_{1})\setminus B|$, which are adjacent to every vertex of the clique. We first label the vertices of the clique, hence, the $A_{\alpha}$ matrix of $G$ may be expressed as:

Let $X_{1}, X_{2}, \dots, X_{k-1}$ be the linearly independent vectors defined as in (ⅰ). Then, we have

Likewise, $X_{2}, \dots, X_{p-1}$ are the eigenvectors of $A_{\alpha}(G)$ that corresponds to the eigenvalue $\alpha(\omega+\beta)-1$. This proves the result.

Assume that a matrix $M$ has a kind of symmetry and may be put in the form

where $X\in R^{n_{1}\times n_{1}}, \; Y \in R^{n_{1}\times n_{2}}$ and $\mathcal{B}, \mathcal{C}\in R^{n_{2}\times n_{2}}$, so that $n = n_{1}+ \eta n_{2}$, where $\eta$ is the number of copies of $\mathcal{B}$. Then the spectrum of $M$ can be found by the following result.

Lemma 2.2. ^[17] Assume that $M$ is a matrix of the form presented in (2.5), having $\eta \geq 1$ copies of the block matrix $\mathcal{B}$ and $Spec(M)$ is the spectrum of $M$. Then $Spec(M) = Spec(\mathcal{B}-\mathcal{C})^{\eta-1}\cup Spec(M^{\prime})$, whereas $M^{\prime} = \begin{pmatrix}X & \sqrt{\eta}Y \\\sqrt{\eta}Y^T & \mathcal{B}+(\eta-1)\mathcal{C}\end{pmatrix}_{n_{1}+n_{2}},$ and $Spec(\mathcal{B}-\mathcal{C})^{\eta-1}$ represents the set of eigenvalues of matrix $\mathcal{B}-\mathcal{C}$ each with multiplicity $\eta-1$.

All our groups are assumed to be finite with identity element represented by $e$. For notations and definitions, we follow ^[22]. The presentation of the dihedral group $D_{2n}$, $n > 2$, is provided as:

Clearly, the last condition is equivalent to $ab = ba^{-1} = ba^{n-1}$. Similarly, the presentation of the semidihedral group $SD_{8n}$ of order $8n$ and the dicyclic group $Q_{4n}$ having order $4n$ are given by:

and

The center of $\mathcal{G}$, symbolized by $Z(\mathcal{G})$, is specified by:

Any finite cyclic group $\mathcal{G}$ of order n can be written as the group $\mathbb{Z}_{n}$ of integers modulo $n$. Clearly, the commuting graph $G = \mathcal{C}(\mathbb{Z}_{n}, \mathbb{Z}_{n})$ is the complete graph $K_{n}$, as every element of $\mathbb{Z}_{n}$ commutes with every other element. The $A_{\alpha}$ spectrum of $\mathcal{C}(\mathbb{Z}_{n}, \mathbb{Z}_{n})$ is $\{(\alpha n-1)^{[n-1]}, n-1\}$, where $[n-1]$ represents the algebraic multiplicity of eigenvalue. It easily follows that $Z(D_{2n}) = \big\{e, a^{\frac{n}{2}}\big\}$, for even $n$ and $Z(D_{2n}) = \{e\}$, for odd $n$. Also, the center of the dicyclic group is $Z(Q_{4n}) = \{e, a^{n}\}$. For the commuting graph ^[4] $G = \mathcal{C}(D_{2n}, Z(D_{2n}))$, G is $K_{1}$, for odd $n$ and $G$ is $K_{2}$, for even $n.$ So, the commuting graphs $\mathcal{C}(\mathcal{G}, Z(\mathcal{G}))$ have simple structures and it will be interesting to investigate the commuting graphs with the non trivial structures.

The next result can be found in ^[4], which gives the structure of $D_{2n}$, where $X$ is $D_{2n}$ itself.

Lemma 2.3. The commuting graph of the dihedral group $D_{2n}$ is

In the following result, we find the $A_{\alpha}$ eigenvalues of the commuting graphs of the dihedral group.

Theorem 2.4. The following properties hold for the commuting graph $\mathcal{C}(D_{2n}, D_{2n})$ of $D_{2n}$.

(ⅰ) If $n$ is odd, then the $A_{\alpha}$ spectrum of $\mathcal{C}(D_{2n}, D_{2n})$ comprises the eigenvalues $\alpha$ and $\alpha n-1$ having multiplicities $n-1$, $n-2$, respectively, and the three zeros of polynomial (2.6).

(ⅱ) If $n$ is even, then the $A_{\alpha}$ spectrum of $\mathcal{C}(D_{2n}, D_{2n})$ comprises the simple eigenvalue $2 \alpha n-1$, the eigenvalue $2 \alpha+1$ having multiplicity $\frac{n}{2}-1$, the eigenvalues $\alpha n-1$ and $4\alpha-1$ having algebraic multiplicities $n-3$ and $\frac{n}{2}$, respectively, and the three zeros of polynomial (2.7).

Proof. (ⅰ) By Lemma 2.3, the commuting graph of $D_{2n}$ for odd $n$ is $\mathcal{C}(D_{2n}, D_{2n}) = K_{1}\vee \big(K_{n-1}\cup \overline{K}_{n}\big)$. Let $\{v_{1}, v_{2}, \dots, v_{n}, u, u_{1}, u_{2}, \dots, u_{n-1}\}$ be the vertex set of $\mathcal{C}(D_{2n}, D_{2n})$, where $v_{i}$'s are the pendent vertices, $u$ is the vertex of degree $2n-1$ and $u_{i}$'s are the vertices of degree $n-1$. As $v_{i}$'s are independent vertices sharing the vertex $u$, so by Theorem 2.1, we get the $A_{\alpha}$ eigenvalue $\alpha$ with algebraic multiplicity $n-1$. Also, $u_{i}$'s are the vertices of clique sharing the vertex $u$ and by (ⅱ) of Theorem 2.1 with $\beta = 1$, we obtain the $A_{\alpha}$ eigenvalue $\alpha(n-1+1)-1 = \alpha n-1$ with algebraic multiplicity $n-2$. In this way, we get $2n-3$ $A_{\alpha}$ eigenvalues and the remaining three $A_{\alpha}$ eigenvalues of $\mathcal{C}(D_{2n}, D_{2n})$ can be found by using eigenequation (2.1). Let $X$ be the eigenvector of $A_{\alpha}(\mathcal{C}(D_{2n}, D_{2n}))$ with $x_{i} = X(v_{i})$, for $i = 1, 2, 3, \dots, 2n.$ Then, it follows that every component of $X$ that corresponds to every pendent vertex is equal to $x_{1}$, component of $X$ that corresponds to the vertex $u$ is $x_{2}$ and the components of $X$ that corresponds to the vertices $u_{i}$'s is equal to $x_{3}$. Therefore, from the eigenequation $A_{\alpha}X = \lambda X$, we have

and the coefficient matrix for the right side of the above equations is:

The characteristic polynomial of the above matrix is given by:

(ⅱ) If $n$ is even, then by Lemma 2.3, the commuting graph of $D_{2n}$ is $K_{2}\vee \Big(K_{n-2} \cup \frac{n}{2}K_{2} \Big)$. Let $v_{1}, v_{2}, \dots, v_{n-2}, v, u, u_{11}, u_{12}, u_{21}, u_{22}, \dots, u_{\frac{n}{2}1}, u_{\frac{n}{2}2}$ be the vertex labelling of $\mathcal{C}(D_{2n}, D_{2n})$, where $v_{i}$'s are the vertices of $K_{n-2}$, $u$ and $v$ are the degree $2n-1$ vertices and $u_{i1}, u_{i2}, \; i = 1, 2, \dots, \frac{n}{2}$ are the vertices of the degree $3.$ Since $v_{i}$'s form the clique and share the same neighbourhood $\{u, v\}$, so by (ⅱ) of Theorem 2.1, $\alpha(n-2+2)-1 = \alpha n-1$ is the $A_{\alpha}$ eigenvalue of $\mathcal{C}(D_{2n}, D_{2n})$ having multiplicity $n-3$. Also, $\{u, v\}$ are the vertices of $K_{2}$ with $\beta = n-2+2\times\frac{n}{2} = 2(n-1)$ and by Theorem 2.1, is shown here that $2\alpha n-1$ is the simple $A_{\alpha}$ eigenvalue of $\mathcal{C}(D_{2n}, D_{2n})$. Similarly, we see that $4\alpha-1$ is the $A_{\alpha}$ eigenvalue of $\mathcal{C}(D_{2n}, D_{2n})$ having multiplicity $\frac{n}{2}$. The remaining $\frac{n}{2}+2$ $A_{\alpha}$ eigenvalues of $\mathcal{C}(D_{2n}, D_{2n})$ can be obtained by using Equation (2.1). If $X$ is the eigenvector of $A_{\alpha}(\mathcal{C}(D_{2n}, D_{2n}))$, then it is evident that every component of $X$ that corresponds to $v_{i}$'s is equal to $x_{1}$, the components of $X$ that corresponds to $u$ and $v$ is $x_{2}$ and the components of $X$ that corresponds to $u_{i1}$ as well as $u_{i2}$ is equal to $x_{i}+2$, for $i = 1, 2, \dots, \frac{n}{2}$. Thus from eigenequation (2.1), we have:

and the coefficient matrix of the right ride of the above system of equations is

Now, applying Lemma 2.2 to the above matrix with

and note that $\eta = \frac{n}{2}$, we get the $A_{\alpha}$ eigenvalue $2\alpha+1$ with algebraic multiplicity $\frac{n}{2}-1$. The other three $A_{\alpha}$ eigenvalues of $\mathcal{C}(D_{2n}, D_{2n})$ are the eigenvalues of the following matrix:

and its characteristic polynomial is

The next lemma gives the structure of the commuting graph of the semidihedral group $SD_{8n}$.

Lemma 2.5. ^[32] The structure of the commuting graph of $SD_{8n}$ is given as:

In the subsequent result, we find the $A_{\alpha}$ eigenvalues of the commuting graph of the semidihedral group.

Theorem 2.6. For the commuting graph $\mathcal{C}(SD_{8n}, SD_{8n})$ of the semidihedral group $SD_{8n}$, the subsequent properties hold.

(ⅰ) If $n$ is odd, then the $A_{\alpha}$ spectrum of $\mathcal{C}(SD_{8n}, SD_{8n})$ comprises the eigenvalues $4\alpha n-1$, $8\alpha n-1$, $8\alpha-1$, $4\alpha+3$ with algebraic multiplicities $4n-5$, $3$, $3n$, $n-1$, respectively, and the eigenvalues of the following matrix:

(ⅱ) If $n$ is even, then the $A_{\alpha}$ spectrum of $\mathcal{C}(SD_{8n}, SD_{8n})$ comprises the simple eigenvalue $8\alpha n-1$, the eigenvalues $4\alpha n-1$, $4\alpha-1$, $2\alpha+1$ having algebraic multiplicities $4n-3$, $2n$, $2n-1$, respectively, and the three eigenvalues of the sequel matrix:

Proof. By using Theorem 2.1, Lemmas 2.2 and 2.5 and proceeding as in (ⅱ) of Theorem 2.4, this result can be proved.

In the next result, we will determine the $A_{\alpha}$ eigenvalues of the commuting graph of the dicyclic group $Q_{4n}.$

Theorem 2.7. The $A_{\alpha}$ spectrum of $\mathcal{C}(Q_{4n}, Q_{4n})$ consists of the simple eigenvalue $4\alpha n-1$, and the eigenvalues $2\alpha n-1$, $4\alpha-1$, $2\alpha+1$ with algebraic multiplicities $2n-3$, $n$, $n-1$, respectively, and the rest three eigenvalues are the zeros of polynomial (2.8).

Proof. The commuting graph $\mathcal{C}(Q_{4n}, Q_{4n})$ ^[5] of $Q_{4n}$ is $\mathcal{C}(Q_{4n}, Q_{4n}) = K_{2}\vee \big(K_{2n-2} \cup n K_{2} \big)$. Let $v_{1}, v_{2}, \dots, v_{2n-2}, v, u, u_{11}, u_{12}, u_{21}, u_{22}, \dots, u_{n1}, u_{n2}$ be the vertex indexing of $\mathcal{C}(Q_{4n}, Q_{4n})$, where $v_{i}$'s are the vertices of the degree $2n-1$, $v$ and $u$ are the vertices of the degree $4n-1$ and $u_{i1}, u_{i2}$ are the vertices of degree $3$, for $i = 1, 2, \dots, n.$ Since $v_{i}$'s form the clique and share the same neighbourhood $\{u, v\}$, so by Theorem 2.1, we have $2n\alpha-1$ is the $A_{\alpha}$ eigenvalue of $\mathcal{C}(Q_{4n}, Q_{4n})$ with algebraic multiplicity $2n-3$. Likewise, $u$ and $v$ form the clique $K_{2}$ and share the same neighbourhood with $\beta = 2n-2+2n = 4n-2$ and again using Theorem 2.1, we obtain the simple $A_{\alpha}$ eigenvalue $4\alpha n-1$. Similarly, for $i = 1, 2, \dots, n$ considering the vertices $u_{i1}$ and $u_{i2}$ with their neighbourhood $\{u, v\}$, we obtain the $A_{\alpha}$ eigenvalue $4\alpha-1$ of $\mathcal{C}(Q_{4n}, Q_{4n})$ with algebraic multiplicity $n$. The other $n+2$, $A_{\alpha}$ eigenvalues of $\mathcal{C}(Q_{4n}, Q_{4n})$ can be found by using Eq (2.1). If $X$ is the eigenvector of $A_{\alpha}(\mathcal{C}(Q_{4n}, Q_{4n}))$, then it is clear that every component of $X$ corresponding to $v_{i}$'s is equal to $x_{1}$, the components of $X$ corresponding to $u$ and $v$ is $x_{2}$ and the components of $X$ corresponding to $u_{i1}$ and $u_{i2}$ is equal to $x_{i}+2$, for $i = 1, 2, \dots, n$. Therefore, by eigenequation (2.1), we have

and the coefficient matrix of the right side of the above system of equations is

Now, applying Lemma 2.2 to the above matrix with

and note that $\eta = n$, we obtain the $A_{\alpha}$ eigenvalue $2\alpha+1$ with algebraic multiplicity $n-1$. The other three $A_{\alpha}$ eigenvalues of $\mathcal{C}(Q_{4n}, Q_{4n})$ are the eigenvalues of the subsequent matrix:

and its characteristic polynomial is given as:

As $A_{\alpha}$ matrix merges the spectral theories of the adjacency matrix, the Laplacian matrix, and the signless Laplacian matrix. Thus for $\alpha = 0$, we find the adjacency spectrum of the commuting graphs of $D_{2n}, SD_{8n}$ and $Q_{4n}$ as already obtained by ^[5,13], by using different techniques. Similarly, for $\alpha = \frac{1}{2}$, we have $A_{\frac{1}{2}}(G) = \frac{1}{2}Q(G)$, so we get the signless Laplacian spectrum of $SD_{8n}$, previously obtained in ^[32], but there is an error in the eigenvalues and with their multiplicities. Also, using the fact that $A_{\alpha_{1}}(G)-A_{\alpha_{2}}(G) = (\alpha_{1}-\alpha_{2})L(G)$, we can find the Laplacian spectrum of the commuting graphs of $D_{2n}, SD_{8n}$ and $Q_{4n}.$

Theorem 2.8. Suppose $\mathcal{C}(\mathcal{G})$ is a commuting graph of a finite group $\mathcal{G}$ and $\sigma(\mathcal{G})$ be its Laplacian spectrum.Then the following hold.

(ⅰ) The Laplacian spectrum of $\mathcal{C}(D_{2n}, D_{2n})$ is

(ⅱ) The Laplacian spectrum of $\mathcal{C}(SD_{8n}, D_{8n})$ is

(ⅲ) The Laplacian spectrum of $\mathcal{C}(Q_{4n}, Q_{4n})$ is

A matrix $M\in M_{n}(\mathbb{F})$ over the field $\mathbb{F}$ is called the integral if its spectrum consists of only integers. Similarly, the Laplacian matrix $L(G)$ of $G$ is integral if all the eigenvalues of $L(G)$ are integers. Next, we have the immediate consequence of Theorem 2.8 about the Laplacian integral graphs.

Theorem 2.9. The commuting graphs of the dihedral group, the semidihedral group and the dicyclic group are Laplacian integral graphs.

3.   Bounds for the $A_{\alpha}$ spectral radius of commuting graphs of non-abelian groups

If a matrix has the special type of symmetry, so that its block representation can be written as:

the rows and the columns of $M$ are partitioned according to a partition $\mathcal{P} = \{ \mathcal{S}_{1}, \mathcal{S}_{2}, \dots, \mathcal{S}_{l}\}$ of the set $S = \{1, 2, \dots, d\}.$ The quotient matrix $\mathcal{Q}$ (see ^[14]) of $M$ is a $d \times d$ matrix whose entries are the average column (row) sums of the blocks $\mathcal{M}_{i, j}$ of $M$. The partition $\mathcal{P}$ is known as regular (equitable) if every block $\mathcal{M}_{i, j}$ of $M$ has the constant column (row) sum and in such case $\mathcal{Q}$ is called the regular quotient matrix. Generally, the eigenvalues of $\mathcal{Q}$ interlace the eigenvalues of $M$. However, for the regular partition $\mathcal{P}$ of $S$, any eigenvalue of the matrix $\mathcal{Q}$ is the eigenvalue of the matrix $M.$

Next, we state a result which is crucial in establishing bounds for the $A_{\alpha}$ spectral radius.

Theorem 3.1. ^[19] Assume that $M_{1}$ and $M_{2}$ are the Hermitian matrices of order $n$ such that $M_{3} = M_{1}+M_{2}$ and $\lambda_{1}(M_{i})\geq \lambda_{2}(M_{i})\geq \dots \geq \lambda_{n}(M_{i}), \; i = 1, 2, 3$ be their eigenvalues. Then

where $\lambda_i$ is the $i$-th largest eigenvalue.Both the inequalities are equalities ^[31] iff there exists a unit vector which is the eigenvector to every of the three eigenvalues involved.

The following result is a consequence of Theorem 3.1 and this can be found in ^[14].

Corollary 3.2. ^[14] Let $M \in M^{*}$ be such that $M = \begin{pmatrix}A & C\\C^{T} & B\end{pmatrix},$ and $\lambda_{n}(M)$ and $\lambda_{1}(M)$ be the smallest and the largest eigenvalues of $M,$ respectively. Then

Now, we give the bounds for the $A_{\alpha}$ eigenvalues of commuting graphs of non-abelian groups.

Theorem 3.3. Let $\lambda_{1}^{\alpha}$ be the $A_{\alpha}$ spectral radius of $\mathcal{C}(D_{2n}, D_{2n})$. Then

Proof. For odd $n$, let $\{u, v_{1}, v_{2}, \dots, v_{n-1}, u_{1}, u_{2}, \dots, u_{n}\}$ be the vertices of $\mathcal{C}(D_{2n}, D_{2n})$, where $u$ is the vertex of degree $2n-1,$ $v_{i}$'s are the vertices of degree $n-1$ and $u_{i}$'s are pendent vertices. Under this labelling, the $A_{\alpha}$ matrix of $\mathcal{C}(D_{2n}, D_{2n})$ is $A_{\alpha}(\mathcal{C}(D_{2n}, D_{2n})) = A + B,$ where block representation of $A$ is

and its regular quotient matrix is

The eigenvalues of $Q$ are $\big\{\alpha, \frac{1}{2} \left(2 \alpha n+n-2\pm\sqrt{-4 \alpha ^2n+4 \alpha n +4 \alpha ^2 n^{2}-4 \alpha n^{2}+n^{2}}\right)\big\}.$

Also, the matrix $B$ is

where $J$ is the matrix of all ones. The regular quotient matrix of $B$ is

and its eigenvalues are $\big\{0, \pm\sqrt{n}(1-\alpha)\big\}.$ Therefore, by Theorem 3.1, the inequality

implies that

For even $n$, with vertex labelling as in Theorem 2.4, the $A_{\alpha}$ matrix of $\mathcal{C}(D_{2n}, D_{2n})$ can be put as $A_{\alpha}(\mathcal{C}(D_{2n}, D_{2n})) = A+B,$ where

and the regular quotient matrix of $A$ is

Now, by Lemma 2.2, with $X = \begin{pmatrix} n-3+2\alpha & 2(1-\alpha)\\ (n-2)(1-\alpha) & \alpha(2n-2)+1 \end{pmatrix}, \; Y = (0), \; \mathcal{B} = (2\alpha+1)$ and $\mathcal{C} = (0)$, we get the eigenvalue $2\alpha+1$ with algebraic multiplicity $n-1$ and the other three eigenvalues of $Q$ are the eigenvalues of the sequel matrix:

The eigenvalues of $M^{\prime}$ are $\big\{2\alpha+1, \frac{1}{2} \left(2\alpha n+n-2\pm\sqrt{n^{2}+8n\alpha-4n^{2}\alpha-8n\alpha^{2}+4 n^{2}\alpha^2}\right)\big\}.$

Similarly,

and its quotient matrix is $\begin{pmatrix} 0 & 0 & 0\\ 0 & 0 & n(1-\alpha)\\ 0 & 2(1-\alpha) & 0 \end{pmatrix}$, whose eigenvalues are $\{0, \sqrt{2n}(1-\alpha)\}$. Therefore, by Theorem 3.1, we obtain

Likewise to Theorem 3.3, we have the subsequent results for the commuting graphs of $SD_{8n}$ and $Q_{4n}$.

Theorem 3.4. Let $\lambda_{1}^{\alpha}$ be the $A_{\alpha}$ spectral radius of the commuting graph $\mathcal{C}(SD_{8n}, SD_{8n})$. Then

Theorem 3.5. Let $\lambda_{1}^{\alpha}$ be the $A_{\alpha}$ spectral radius of $\mathcal{C}(Q_{4n}, Q_{4n})$. Then

Finally, we obtain the upper bounds for the $A_{\alpha}$ spectral radius and the least $A_{\alpha}$ eigenvalue of commuting graphs of non-abelian groups.

Theorem 3.6. Let $\lambda_{1}^{\alpha}$ and $\lambda_{n}^{\alpha}$ be the $A_{\alpha}$ spectral radius and the smallest $A_{\alpha}$ eigenvalue of the commuting graph $\mathcal{C}(D_{2n}, D_{2n})$. Then

Proof. Labelling the vertices as in Theorem 2.4, the $A_{\alpha}$ matrix of $\mathcal{C}(D_{2n}, D_{2n})$ for odd $n$ can be written as $A_{\alpha}(\mathcal{C}(D_{2n}, D_{2n})) = \begin{pmatrix} A_{n+1} & C_{(n+1)\times (n-1)}\\ C^{T} & B_{n-1} \end{pmatrix},$ where

and

By applying Lemma 2.2 to the matrix $B$, with $X = (0), Y = (0), \mathcal{B} = \alpha(n-1),$ and $\mathcal{C} = (1-\alpha)$. Consequently, $\alpha n-1$ and $n+\alpha-2$ are its only distinct eigenvalues. Also, the regular quotient matrix of $A$ is

and its eigenvalues are $\alpha n\pm \sqrt{\alpha^{2}(n^{2}-n+1)+n(1-2\alpha)}.$ Therefore, by Corollary 3.2, we have

For even $n$, indexing the vertices as in Theorem 2.4, the $A_{\alpha}$ matrix of $\mathcal{C}(D_{2n}, D_{2n})$ can be written as $A_{\alpha}(\mathcal{C}(D_{2n}, D_{2n})) = \begin{pmatrix} A_{n} & C_{n}\\ C^{T} & B_{n} \end{pmatrix},$ where

Now, the quotient matrix of $A$ is $\begin{pmatrix} n-3+2\alpha & 2(1-\alpha)\\ (n-2)(1-\alpha) & \alpha(2n-2)+1 \end{pmatrix}$ and its eigenvalues are $\frac{1}{2} \left(2 \alpha n+n-2\pm \sqrt{n(-8 \alpha ^2+8 \alpha +4 \alpha ^2 n-4 \alpha n+n)}\right).$ Similarly, the regular quotient of $B$ is

and we have $2\alpha +1$ is the eigenvalue of $Q$ with multiplicity $\frac{n}{2}$. Thus, by Corollary 3.2, we obtain

Following the proof of Theorem 3.6, we have the similar results for the commuting graphs of the semidihedral and the dicyclic groups.

Theorem 3.7. Let $\lambda_{1}^{\alpha}$ and $\lambda_{n}^{\alpha}$ be the $A_{\alpha}$ spectral radius and the smallest $A_{\alpha}$ eigenvalue of the commuting graph $\mathcal{C}(SD_{8n}, SD_{8n})$. Then

Theorem 3.8. Let $\lambda_{1}^{\alpha}$ and $\lambda_{n}^{\alpha}$ be the $A_{\alpha}$ spectral radius and the smallest $A_{\alpha}$ eigenvalue of the commuting graph $\mathcal{C}(Q_{4n}, Q_{4n})$. Then

4.   Conclusions

In this article, the adjacency eigenvalues, the Laplacian eigenvalues, the signless Laplacian eigenvalues, and the generalized adjacency eigenvalues of graphs are given, including the bounds on the smallest and largest eigenvalues. The $A_{\alpha}$ matrix makes it very interesting to study the eigenvalues of well-known matrices in a very natural setting. Spectral properties of the graph defined by algebraic structures (groups, rings, modules, vector spaces, and others) have attracted many researchers, and various interesting problems have been solved both in combinatorics and algebra; for some recent developments, see ^{[2,5,6,7,8,9,10,26,27,28,29,30,32]}. However, the $A_{\alpha}$ spectrum of all commuting and non-commuting graphs of groups remains open at large.

Acknowledgments

This research was supported by Taif University Researchers Supporting Project number (TURSP-2020/144), Taif University, Taif, Saudi Arabia.

The authors would like to gratefully acknowledge anonymous referee's constructive comments and suggestions, which greatly helped to improve the readability of the manuscript.

Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.