Signless Laplacian spectrum of power graph of certain finite non-commutative groups

Banerjee, Subarsha

doi:10.2478/ijmce-2026-0003

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1

Introduction

In [1], Kelarev and Quinn began studying the power graph of a finite semigroup. The directed power graph of a semigroup S, 𝒫(S), has vertex set as S. There exists a directed edge from u to v if v is some power of u. The undirected power graph 𝒫(G) of a finite group G was introduced in [2]. 𝒫(G) has vertex set as G, and two vertices u and v are adjacent if and only if u = v^m or v = uⁿ for some m, n. From [2], it is known that 𝒫(G) is always connected and it is complete if and only if G is a cyclic group of order 1 or p^m where p is a prime number and m is a positive integer. In [3], it was proved that non-isomorphic finite groups may have isomorphic power graphs, but finite abelian groups with isomorphic power graphs must be isomorphic. Moreover, it was conjectured that two finite groups with isomorphic power graphs must have the same number of elements of each order. In [4], the conjecture was proved. In [5], several structural properties of power graph of finite groups have been studied. Various interesting results on the power graph of finite groups and semigroups have been studied in [6].

Given a simple graph 𝒢, the signless Laplacian matrix Q(𝒢) of 𝒢 is defined as Q(𝒢) = Deg (𝒢)+A(𝒢). Here, A(𝒢) is the adjacency matrix of 𝒢, and Deg (𝒢) is the diagonal matrix of vertex degrees. The signless Laplacian matrix (see [7–9]) Q(𝒢) has all its eigenvalues as non-negative. The set of eigenvalues of the adjacency matrix and signless Laplacian matrix of 𝒢 respectively form the adjacency, and signless Laplacian spectrum of 𝒢. Let λ₁ ≤ λ₂ ≤ ⋯ ≤ λ_k denote the eigenvalues of Q(𝒢) having multiplicities m₁, m₂, ⋯, m_k. Then the spectrum of Q(𝒢), denoted by σ(𝒢), is described as follows: $σ (𝒢) = (\begin{matrix} λ_{1} & λ_{2} & \dots & λ_{k} \\ m_{1} & m_{2} & \dots & m_{k} \end{matrix})$ \sigma ({\cal G}) = \left( {\matrix{ {{\lambda _1}} & {{\lambda _2}} & \cdots & {{\lambda _k}} \cr {{m_1}} & {{m_2}} & \cdots & {{m_k}} \cr } } \right). Thus, $σ (K_{n}) = (\begin{matrix} 2 (n - 1) & n - 2 \\ 1 & n - 1 \end{matrix})$ \sigma ({K_n}) = \left( {\matrix{ {2(n - 1)} & {n - 2} \cr 1 & {n - 1} \cr } } \right) for n ≥ 2 where, K_n is the complete graph. The largest eigenvalue of Q is known as the Q-spectral radius of G. We shall denote the finite cyclic group by ℤ_n, the dihedral group by D_n, the dicyclic group by Dic_4n, and the generalized quaternion group by Q_2^k+2. The Laplacian spectrum of 𝒫(ℤ_n) and 𝒫(D_n) was studied in [10]. Various other structural properties of power graph of finite groups have been studied in [11–13]. For the past few years, scholars have interested in studying the spectrum of graphs connected to algebraic structures in [14–19]. The adjacency spectra of power graph of various finite groups have been studied in [20]. In [21], Banerjee and Adhikari initiated the study of eigenvalues of the signless Laplacian matrix of 𝒫(ℤ_n). A recent survey on power graph of finite groups has been studied in [22].

The above works motivate us to calculate the signless Laplacian spectrum of the power graph of certain finite non-commutative groups. We compute the signless Laplacian spectrum of the power graph of elementary abelian groups of prime power order, Mathieu group M₁₁, and Q_2^k+2. Finally, we provide bounds on the signless Laplacian spectral radius of 𝒫(Dic_4n). In [23] and [24], some definitions of standard terms related to graph theory and group theory used in this paper have been given. Throughout the paper, the complement of a graph 𝒢 is denoted by $\bar{𝒢}$ {\bar {\cal G}}. The characteristic polynomial of a matrix M is denoted by Θ(M;x).

The rest of this paper is organized as follows: In Section 2, we present some definitions and theorems. In Section 3, we provide some applications of studying the signless Laplacian spectrum of power graph of various finite groups in brief. We determine the eigenvalues of the signless Laplacian matrix of the power graph of elementary abelian group of order pⁿ, and the smallest sporadic group M₁₁, and the generalized quaternion group Q_2^k+2. Moreover, we provide lower and upper bounds on the largest eigenvalue of the signless Laplacian matrix of dicyclic group Dic_n. In Section 4, we introduce the main contribution of the paper and also discuss some related work which can be considered for future research.

2

Preliminaries

In this section of the paper, we present some theorems and definitions used in this paper.

Definition 1.

If G is a commutative group and all elements of G other than the identity element have the same order, then G is said to be an elementary abelian group.

If G is an elementary abelian group with identity element e, then for any element g(≠e) of G, the order of g will be p for some prime number.

Theorem 1.

Suppose G is an elementary abelian group having order pⁿ. If $ℓ = \frac{p^{n} - 1}{p - 1}$ \ell = {{{p^n} - 1} \over {p - 1}}, then the signless Laplacian spectrum of 𝒫(G) is given as follows: $σ (𝒫 (G)) = (\begin{matrix} 2 p - 3 & \frac{ℓ a^{2} + t + \sqrt{a^{4} ℓ^{2} - 2 a^{2} ℓ t + 4 a^{2} ℓ + t^{2}}}{2} & \frac{ℓ a^{2} + t - \sqrt{a^{4} ℓ^{2} - 2 a^{2} ℓ t + 4 a^{2} ℓ + t^{2}}}{2} \\ ℓ - 1 & 1 & 1 \end{matrix}) .$ \sigma (\mid {\cal P}(G)) = \left( {\matrix{ {2p - 3} & {{{\ell {a^2} + t + \sqrt {{a^4}{\ell ^2} - 2{a^2}\ell t + 4{a^2}\ell + {t^2}} } \over 2}} & {{{\ell {a^2} + t - \sqrt {{a^4}{\ell ^2} - 2{a^2}\ell t + 4{a^2}\ell + {t^2}} } \over 2}} \cr {\ell - 1} & 1 & 1 \cr } } \right).

Here, $a = \sqrt{p - 1}$ a = \sqrt {p - 1} , t = 2p – 3.

Proof.

Given $ℓ = \frac{p^{n} - 1}{p - 1}$ \ell = {{{p^n} - 1} \over {p - 1}}, using [25, Theorem 8], we find that 𝒫(G) is isomorphic to $𝒢 [K_{1}, \underset{ℓ times}{\underset{︸}{K_{p - 1}, K_{p - 1}, \dots, K_{p - 1}}}],$ {\cal G}[{K_1},\underbrace {{K_{p - 1}},{K_{p - 1}}, \cdots ,{K_{p - 1}}}_{\ell {\rm{ times }}}],, where $𝒢 = K_{1} + {\bar{K}}_{ℓ}$ {\cal G} = {K_1} + {{\bar K}_\ell }. We note that 𝒢 has ℓ + 1 vertices. If we denote the vertices of 𝒢 by {1,2,3,4, ⋯,ℓ,ℓ + 1}, then 𝒢 looks like in Figure 1.

Definition 2.

[26, Page 59] The dicyclic group of order 4n, denoted by Dic_4n, is represented as follows: ${Dic}_{4 n} = 〈 a, b : a^{2 n} = e, a^{n} = b^{2}, a b = b a^{- 1} 〉 .$ {\rm{Di}}{{\rm{c}}_{4n}} = \langle a,b:{a^{2n}} = e,{a^n} = {b^2},ab = b{a^{ - 1}}\rangle .

If n = 2, then Dic_4n is known as the quaternion group. Moreover, if n = 2^k for some natural number k, then Dic_4n is known as the generalized quaternion group, and it is denoted by Q_2^k+2.

3

Applications

The signless Laplacian eigenvalues of a graph are associated to the graphs structural properties. In the case of graphs derived from algebraic structures such as power graph which has been studied in this paper, spectral analysis with regard to signless Laplacian matrix can help address a variety of algebraic structure-related concerns. Certain invariants related to graph connectivity and structure can also be derived from the signless Laplacian spectrum. The signless Laplacian spectrum can be used to study partitioning and cut problems in power graph of finite groups, where we are interested in separating a graph into subgraphs with minimal edge cuts.

3.1

Signless Laplacian spectrum of power graph of finite groups

Here, we shall determine the signless Laplacian spectrum of power graph of elementary abelian group, the smallest sporadic group M₁₁, and the generalized quaternion group Q_2^k+2. From Definition 1 and Theorem 1, we note that N_𝒢(i) = {1} for all 2 ≤ i ≤ ℓ + 1. Moreover, N_𝒢(1) = {2,3,4, ⋯,ℓ,ℓ+1}. Thus, we observe that $N_{1} = \sum_{i = 2}^{ℓ + 1} (p - 1) = ℓ (p - 1)$ {N_1} = \sum\nolimits_{i = 2}^{\ell + 1} {(p - 1)} = \ell (p - 1). Moreover, N_i = 1 for 2 ≤ i ≤ ℓ + 1. We further note that K_p−1 is a p – 2 regular graph whose signless Laplacian eigenvalues are 2(p – 2) having multiplicity 1, and p – 3 having multiplicity p – 2.

Hence, $σ (K_{p - 1}) = (\begin{matrix} 2 (p - 2) & p - 3 \\ 1 & p - 2 \end{matrix})$ \sigma \left( {{K_{p - 1}}} \right) = \left( {\matrix{ {2(p - 2)} & {p - 3} \cr 1 & {p - 2} \cr } } \right). Also, $σ (K_{1}) = (\begin{matrix} 0 \\ 1 \end{matrix})$ \sigma \left( {{K_1}} \right) = \left( \matrix{ 0 \cr 1 \cr} \right).

Using [27, Theorem 2.₁] and the above information, the signless Laplacian spectrum of 𝒫(G) consists of p – 2 having multiplicity ℓ(p – 2), and remaining eigenvalues are contained in the spectrum of C_𝒫(G) where, $C_{𝒫(G)} = {(\begin{matrix} ℓ (p - 1) & \sqrt{p - 1} & \sqrt{p - 1} & \sqrt{p - 1} & \dots & \dots & \sqrt{p - 1} & \sqrt{p - 1} \\ \sqrt{p - 1} & 2 p - 3 & 0 & 0 & \dots & \dots & 0 & 0 \\ \sqrt{p - 1} & 0 & 2 p - 3 & 0 & \dots & \dots & 0 & 0 \\ \sqrt{p - 1} & 0 & 0 & 2 p - 3 & \dots & \dots & 0 & 0 \\ \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ c d o t s \\ \sqrt{p - 1} & 0 & 0 & 0 & \dots & \dots & 2 p - 3 & 0 \\ \sqrt{p - 1} & 0 & 0 & 0 & \dots & \dots & 0 & 2 p - 3 \end{matrix})}_{(ℓ + 1) \times (ℓ + 1)} .$ {C_{{\cal P}(G)}} = {\left( {\matrix{ {\ell (p - 1)} & {\sqrt {p - 1} } & {\sqrt {p - 1} } & {\sqrt {p - 1} } & \cdots & \cdots & {\sqrt {p - 1} } & {\sqrt {p - 1} } \cr {\sqrt {p - 1} } & {2p - 3} & 0 & 0 & \cdots & \cdots & 0 & 0 \cr {\sqrt {p - 1} } & 0 & {2p - 3} & 0 & \cdots & \cdots & 0 & 0 \cr {\sqrt {p - 1} } & 0 & 0 & {2p - 3} & \cdots & \cdots & 0 & 0 \cr \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \cr \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \cr \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & {} \cr {cdots} & {} & {} & {} & {} & {} & {} & {} \cr {\sqrt {p - 1} } & 0 & 0 & 0 & \cdots & \cdots & {2p - 3} & 0 \cr {\sqrt {p - 1} } & 0 & 0 & 0 & \cdots & \cdots & 0 & {2p - 3} \cr } } \right)_{(\ell + 1) \times (\ell + 1)}}.

For the sake of convenience, we shall write matrix C_𝒫(G) as follows: $C_{𝒫 (G)} = {(\begin{matrix} ℓ a^{2} & a & a & a & \dots & \dots & a & a \\ a & t & 0 & 0 & \dots & \dots & 0 & 0 \\ a & 0 & t & 0 & \dots & \dots & 0 & 0 \\ a & 0 & 0 & t & \dots & \dots & 0 & 0 \\ \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ a & 0 & 0 & 0 & \dots & \dots & t & 0 \\ a & 0 & 0 & 0 & \dots & \dots & 0 & t \end{matrix})}_{(ℓ + 1) \times (ℓ + 1)},$ {C_{{\cal P}(G)}} = {\left( {\matrix{ {\ell {a^2}} & a & a & a & \cdots & \cdots & a & a \cr a & t & 0 & 0 & \cdots & \cdots & 0 & 0 \cr a & 0 & t & 0 & \cdots & \cdots & 0 & 0 \cr a & 0 & 0 & t & \cdots & \cdots & 0 & 0 \cr \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \cr \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \cr \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \cr a & 0 & 0 & 0 & \cdots & \cdots & t & 0 \cr a & 0 & 0 & 0 & \cdots & \cdots & 0 & t \cr } } \right)_{(\ell + 1) \times (\ell + 1)}}, where $a = \sqrt{p - 1}$ a = \sqrt {p - 1} and t = 2p – 3. We shall now find the characteristic polynomial of C_𝒫(G). $\begin{array}{l} Θ (C_{𝒫 (G)}; x) & = \det (x I - C_{𝒫 (G)}) \\ = \det {(\begin{matrix} x - ℓ a^{2} & - a & - a & - a & \dots & \dots & - a & - a \\ - a & x - t & 0 & 0 & \dots & \dots & 0 & 0 \\ - a & 0 & x - t & 0 & \dots & \dots & 0 & 0 \\ - a & 0 & 0 & x - t & \dots & \dots & 0 & 0 \\ \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ - a & 0 & 0 & 0 & \dots & \dots & x - t & 0 \\ - a & 0 & 0 & 0 & \dots & \dots & 0 & x - t \end{matrix})}_{(ℓ + 1) \times (ℓ + 1)} . \end{array}$ \matrix{ {\Theta ({C_{(G)}};x)} \hfill & { = \det (xI - {C_{(G)}})} \hfill \cr {} \hfill & { = \det {{(\matrix{ {x - \ell {a^2}} & { - a} & { - a} & { - a} & \cdots & \cdots & { - a} & { - a} \cr { - a} & {x - t} & 0 & 0 & \cdots & \cdots & 0 & 0 \cr { - a} & 0 & {x - t} & 0 & \cdots & \cdots & 0 & 0 \cr { - a} & 0 & 0 & {x - t} & \cdots & \cdots & 0 & 0 \cr \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \cr \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \cr \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \cr { - a} & 0 & 0 & 0 & \cdots & \cdots & {x - t} & 0 \cr { - a} & 0 & 0 & 0 & \cdots & \cdots & 0 & {x - t} \cr } )}_{(\ell + 1) \times (\ell + 1)}}.} \hfill \cr }

We shall evaluate the above determinant using an inductive process. We define ℳ_{(ℓ+1)×(ℓ+1)} = (xI – C_𝒫(G))_{(ℓ+1)×(ℓ+1)}. Moreover, we shall obtain ℳ_ℓ×ℓ from ℳ_{(ℓ+1)×(ℓ+1)} by deleting the (ℓ + 1)^th row and column. Thus, $ℳ_{ℓ \times ℓ} = {(\begin{matrix} x - ℓ a^{2} & - a & - a & - a & \dots & \dots & \dots & - a \\ - a & x - t & 0 & 0 & \dots & \dots & \dots & 0 \\ - a & 0 & x - t & 0 & \dots & \dots & \dots & 0 \\ - a & 0 & 0 & x - t & \dots & \dots & \dots & 0 \\ \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ - a & 0 & 0 & 0 & \dots & \dots & \dots & x - t \end{matrix})}_{ℓ \times ℓ},$ {{\cal M}_{\ell \times \ell }} = {\left( {\matrix{ {x - \ell {a^2}} & { - a} & { - a} & { - a} & \cdots & \cdots & \cdots & { - a} \cr { - a} & {x - t} & 0 & 0 & \cdots & \cdots & \cdots & 0 \cr { - a} & 0 & {x - t} & 0 & \cdots & \cdots & \cdots & 0 \cr { - a} & 0 & 0 & {x - t} & \cdots & \cdots & \cdots & 0 \cr \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \cr \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \cr \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \cr { - a} & 0 & 0 & 0 & \cdots & \cdots & \cdots & {x - t} \cr } } \right)_{\ell \times \ell }}, and $ℳ_{2 \times 2} = {(\begin{matrix} x - ℓ a^{2} & - a \\ - a & x - t \end{matrix})}_{2 \times 2} .$ {{\cal M}_{2 \times 2}} = {\left( {\matrix{ {x - \ell {a^2}} & { - a} \cr { - a} & {x - t} \cr } } \right)_{2 \times 2}}.

The matrices ℳ_{(ℓ–i)×(ℓ–i)} for 1 ≤ i ≤ ℓ – 2 can be obtained along similar lines as shown above.

Now, expanding ℳ_{(ℓ+1)×(ℓ+1)} along the (ℓ +1)^th row, we obtain, 1 $\det (ℳ_{(ℓ + 1) \times (ℓ + 1)}) = (x - t) \det (ℳ_{ℓ \times ℓ}) - a^{2} {(x - t)}^{ℓ - 1} .$ \det ({{\cal M}_{(\ell + 1) \times (\ell + 1)}}) = (x - t)\det ({{\cal M}_{\ell \times \ell }}) - {a^2}{(x - t)^{\ell - 1}}.

In order to evaluate det(ℳ_ℓ×ℓ), we shall expand along the ℓ^th row of ℳ_ℓ×ℓ. We observe, 2 $\det (ℳ_{ℓ \times ℓ}) = (x - t) \det (ℳ_{(ℓ - 1) \times (ℓ - 1)}) - a^{2} (x - t)^{ℓ - 2} .$ \det ({{\cal M}_{\ell \times \ell }}) = (x - t)\det ({{\cal M}_{(\ell - 1) \times (\ell - 1)}}) - {a^2}{(x - t)^{\ell - 2}}.

On using equation (2) into equation (1), we obtain, 3 $\det (ℳ_{(ℓ + 1) \times (ℓ + 1)}) = (^{x - t) 2} \det (ℳ_{(ℓ - 1) \times (ℓ - 1)}) - 2 a^{2} {(x - t)}^{ℓ - 1} .$ \det ({{\cal M}_{(\ell + 1) \times (\ell + 1)}}) = {(x - t)^2}\det ({{\cal M}_{(\ell - 1) \times (\ell - 1)}}) - 2{a^2}{(x - t)^{\ell - 1}}.

If we continue the above process, we shall obtain $\begin{array}{l} \det (ℳ_{(ℓ + 1) \times (ℓ + 1)}) & = {(x - t)}^{ℓ - 1} \det (ℳ_{2 \times 2}) - a^{2} (ℓ - 1) {(x - t)}^{ℓ - 1} \\ = {(x - t)}^{ℓ - 1} (\det (ℳ_{2 \times 2}) - a^{2} (ℓ - 1)) \\ = {(x - t)}^{ℓ - 1} (x^{2} - x (ℓ a^{2} + t) + t ℓ a^{2} - a^{2} - a^{2} (ℓ - 1)) \\ = {(x - t)}^{ℓ - 1} (x^{2} - x (ℓ a^{2} + t) + a^{2} (t ℓ - 1 - ℓ + 1)) \\ = {(x - t)}^{ℓ - 1} (x^{2} - x (ℓ a^{2} + t) + a^{2} ℓ (t - 1)) . \end{array}$ \matrix{ {\det \left( {{{\cal M}_{(\ell + 1) \times (\ell + 1)}}} \right)} \hfill & { = {{(x - t)}^{\ell - 1}}\det \left( {{{\cal M}_{2 \times 2}}} \right) - {a^2}(\ell - 1){{(x - t)}^{\ell - 1}}} \hfill \cr {} \hfill & { = {{(x - t)}^{\ell - 1}}\left( {\det \left( {{{\cal M}_{2 \times 2}}} \right) - {a^2}(\ell - 1)} \right)} \hfill \cr {} \hfill & { = {{(x - t)}^{\ell - 1}}\left( {{x^2} - x\left( {\ell {a^2} + t} \right) + t\ell {a^2} - {a^2} - {a^2}(\ell - 1)} \right)} \hfill \cr {} \hfill & { = {{(x - t)}^{\ell - 1}}\left( {{x^2} - x\left( {\ell {a^2} + t} \right) + {a^2}(t\ell - 1 - \ell + 1)} \right)} \hfill \cr {} \hfill & { = {{(x - t)}^{\ell - 1}}\left( {{x^2} - x\left( {\ell {a^2} + t} \right) + {a^2}\ell (t - 1)} \right).} \hfill \cr }

The roots of x² – x(ℓa²+t) + a²ℓ(t – 1) = 0 are $\frac{ℓ a^{2} + t \pm \sqrt{a^{4} ℓ^{2} - 2 a^{2} ℓ t + 4 a^{2} ℓ + t^{2}}}{2}$ {{\ell {a^2} + t \pm \sqrt {{a^4}{\ell ^2} - 2{a^2}\ell t + 4{a^2}\ell + {t^2}} } \over 2}.

Since the roots of Θ(C_𝒫(G); x) = 0 are 2p – 3 having multiplicity ℓ – 1 and $\frac{ℓ a^{2} + t \pm \sqrt{a^{4} ℓ^{2} - 2 a^{2} ℓ t + 4 a^{2} ℓ + t^{2}}}{2}$ {{\ell {a^2} + t \pm \sqrt {{a^4}{\ell ^2} - 2{a^2}\ell t + 4{a^2}\ell + {t^2}} } \over 2} having multiplicity 1, the result follows. We know that any finite simple group will be either cyclic, or alternating, or it will belong Lie groups, or else it is one of 26 exceptions known as the sporadic groups. Among the 26 sporadic groups, 5 were discovered by the famous mathematician Émile Léonard Mathieu in 1860. The smallest sporadic group M₁₁, known as the Mathieu group of degree 11, has order 7920 = 2⁴ × 3² × 5 × 11. For more information about M₁₁, it has been presented in [28]. The graph 𝒫(M₁₁) has been studied in detail in [29]. The structure of 𝒫(M₁₁) has been depicted in [29, Figure 3].

Theorem 2.

The signless Laplacian spectrum of the proper power graph of the sporadic group M₁₁ is given as follows: $σ (𝒫 (M_{11}) = (\begin{matrix} 0 & 2 & 3 & 5 & 5 & 7 & 8 & 11 & α_{1} & α_{2} & β_{1} & β_{2} & β_{3} & β_{4} \\ 55 & 1243 & 660 & 2970 & 396 & 432 & 785 & 330 & 144 & 144 & 165 & 165 & 165 & 165 \end{matrix}) .$ \sigma ({\cal P}({M_{11}}) = \left( {\matrix{ 0 & 2 & 3 & 5 & 5 & 7 & 8 & {11} & {{\alpha _1}} & {{\alpha _2}} & {{\beta _1}} & {{\beta _2}} & {{\beta _3}} & {{\beta _4}} \cr {55} & {1243} & {660} & {2970} & {396} & {432} & {785} & {330} & {144} & {144} & {165} & {165} & {165} & {165} \cr } } \right).

Here, α₁, α₁ are roots of the polynomial equation x² – 21x + 70 = 0, and β₁,β₂,β₃,β₄ are roots of the polynomial equation x⁴ – 52x³ + 849x² – 5110x + 8232 = 0.

Proof.

From [29], it is known that the 𝒫(M₁₁) consists of 165 copies of a graph L, 55 copies of 𝒫(ℤ₃), 396 copies of 𝒫(ℤ₅), and 144 copies of 𝒫(ℤ₁₁). All of them are connected to each other through the identity element of M₁₁. Consequently, the proper power graph of M₁₁ consists of 165 copies of L*, 55 copies of 𝒫(ℤ₂), 396 copies of 𝒫(ℤ₄), and 144 copies of 𝒫(ℤ₁₀). The graph of L* is shown in Figure 2.

Using [21, Theorem 3.1], we have $σ (𝒫 (ℤ_{2})) = (\begin{array}{l} 2 & 0 \\ 1 & 1 \end{array})$ \sigma \left( {{\cal P}\left( {{_2}} \right)} \right) = \left( {\matrix{ 2 \hfill & 0 \hfill \cr 1 \hfill & 1 \hfill \cr } } \right), and $σ (𝒫 (ℤ_{4})) = (\begin{array}{l} 6 & 2 \\ 1 & 3 \end{array})$ \sigma \left( {{\cal P}\left( {{_4}} \right)} \right){\mkern 1mu} = \left( {\matrix{ 6 \hfill & 2 \hfill \cr 1 \hfill & 3 \hfill \cr } } \right). Moreover, using [21, Theorem 5.4], the signless Laplacian spectrum of 𝒫(ℤ₁₀) is given as follows: $σ ((𝒫 ℤ_{10})) = (\begin{matrix} 8 & 7 & p_{1} & p_{2} \\ 5 & 3 & 1 & 1 \end{matrix}),$ \sigma \left( {{\cal P}\left( {{_{10}}} \right)} \right) = \left( {\matrix{ 8 & 7 & {{p_1}} & {{p_2}} \cr 5 & 3 & 1 & 1 \cr } } \right), where p₁, p₂ are roots of the quadratic equation x² – 21x + 70 = 0. Now, we find that L^* = 𝒢_L* [K₆,K₆,K₆,K₁,K₂,K₂,K₂,K₂,K₂], where 𝒢_L* is of the following Figure 3.

We have N_{𝒢_L*}(i) = {4} for 1 ≤ i ≤ 3, N_{𝒢_L*}(i) = {4,9} for 5 ≤ i ≤ 8, N_{𝒢_L*}(4) = {1,2,3,5,6,7,8}, and N_{𝒢_L*}(9) = {5,6,7,8}

Here, n₁ = n₂ = n₃ = 6, n₄ = 1, n₅ = n₆ = n₇ = n₈ = n₉ = 2. Consequently, N_i = 1 for all 1 ≤ i ≤ 3, and N_i = 1 + 2 = 3 for all 5 ≤ i ≤ 8. Moreover, N₄ = 6 + 6 + 6 + 2 + 2 + 2 + 2 = 26, and N₉ = 2 + 2 + 2 + 2 = 8. Now, $σ (K_{6}) = (\begin{matrix} 10 & 4 \\ 1 & 5 \end{matrix})$ \sigma \left( {{K_6}} \right) = \left( {\matrix{ {10} & 4 \cr 1 & 5 \cr } } \right), and $σ (K_{2}) = (\begin{array}{l} 2 & 0 \\ 1 & 1 \end{array})$ \sigma \left( {{K_2}} \right) = \left( {\matrix{ 2 \hfill & 0 \hfill \cr 1 \hfill & 1 \hfill \cr } } \right). Thus, using [27, Theorem 2.1], the signless Laplacian eigenvalues of L* are 5 having multiplicity 15, 3 having multiplicity 4, 8 having multiplicity 1, and rest are contained in the following: $C_{𝒢_{L^{*}}} = {(\begin{matrix} 11 & 0 & 0 & \sqrt{6} & 0 & 0 & 0 & 0 & 0 \\ 0 & 11 & 0 & \sqrt{6} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 11 & \sqrt{6} & 0 & 0 & 0 & 0 & 0 \\ \sqrt{6} & \sqrt{6} & \sqrt{6} & 26 & \sqrt{2} & \sqrt{2} & \sqrt{2} & \sqrt{2} & 0 \\ 0 & 0 & 0 & \sqrt{2} & 5 & 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & \sqrt{2} & 0 & 5 & 0 & 0 & 2 \\ 0 & 0 & 0 & \sqrt{2} & 0 & 0 & 5 & 0 & 2 \\ 0 & 0 & 0 & \sqrt{2} & 0 & 0 & 0 & 5 & 2 \\ 0 & 0 & 0 & 0 & 2 & 2 & 2 & 2 & 10 \end{matrix})}_{9 \times 9} .$ {C_{{{\cal G}_{{L^*}}}}} = {\left( {\matrix{ {11} & 0 & 0 & {\sqrt 6 } & 0 & 0 & 0 & 0 & 0 \cr 0 & {11} & 0 & {\sqrt 6 } & 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & {11} & {\sqrt 6 } & 0 & 0 & 0 & 0 & 0 \cr {\sqrt 6 } & {\sqrt 6 } & {\sqrt 6 } & {26} & {\sqrt 2 } & {\sqrt 2 } & {\sqrt 2 } & {\sqrt 2 } & 0 \cr 0 & 0 & 0 & {\sqrt 2 } & 5 & 0 & 0 & 0 & 2 \cr 0 & 0 & 0 & {\sqrt 2 } & 0 & 5 & 0 & 0 & 2 \cr 0 & 0 & 0 & {\sqrt 2 } & 0 & 0 & 5 & 0 & 2 \cr 0 & 0 & 0 & {\sqrt 2 } & 0 & 0 & 0 & 5 & 2 \cr 0 & 0 & 0 & 0 & 2 & 2 & 2 & 2 & {10} \cr } } \right)_{9 \times 9}}.

We have: 4 $\begin{array}{l} Θ (C_{𝒢_{L^{*}}}, x) & = x^{9} - 89 x^{8} + 3299 x^{7} - 67465 x^{6} + 842381 x^{5} - 6670379 x^{4} \\ + 33500857 x^{3} - 102819755 x^{2} + 174632150 x - 124509000 \\ = {(x - 11)}^{2} {(x - 5)}^{3} (x^{4} - 52 x^{3} + 849 x^{2} - 5110 x + 8232) . \end{array}$ \matrix{ {\Theta \left( {{C_{{{\cal G}_{{L^*}}}}},x} \right)} \hfill & { = {x^9} - 89{x^8} + 3299{x^7} - 67465{x^6} + 842381{x^5} - 6670379{x^4}} \hfill \cr {} \hfill & { + 33500857{x^3} - 102819755{x^2} + 174632150x - 124509000} \hfill \cr {} \hfill & { = {{(x - 11)}^2}{{(x - 5)}^3}\left( {{x^4} - 52{x^3} + 849{x^2} - 5110x + 8232} \right).} \hfill \cr }

Hence, using 4, we obtain 5 $\begin{array}{l} Θ (L^{*}, x) & = {(x - 5)}^{15} \times {(x - 3)}^{4} \times (x - 8) \times Θ (C_{C_{L^{*}}}, x) \\ = {(x - 5)}^{15} {(x - 3)}^{4} (x - 8) \times ({(x - 11)}^{2} {(x - 5)}^{3} (x^{4} - 52 x^{3} + 849 x^{2} - 5110 x + 8232)) \\ = {(x - 3)}^{4} {(x - 5)}^{18} (x - 8) {(x - 11)}^{2} (x^{4} - 52 x^{3} + 849 x^{2} - 5110 x + 8232) . \end{array}$ \matrix{ {\Theta \left( {{L^*},x} \right)} \hfill & { = {{(x - 5)}^{15}} \times {{(x - 3)}^4} \times (x - 8) \times \Theta \left( {{C_{{{\cal C}_{{L^*}}}}},x} \right)} \hfill \cr {} \hfill & { = {{(x - 5)}^{15}}{{(x - 3)}^4}(x - 8) \times \left( {{{(x - 11)}^2}{{(x - 5)}^3}\left( {{x^4} - 52{x^3} + 849{x^2} - 5110x + 8232} \right)} \right)} \hfill \cr {} \hfill & { = {{(x - 3)}^4}{{(x - 5)}^{18}}(x - 8){{(x - 11)}^2}\left( {{x^4} - 52{x^3} + 849{x^2} - 5110x + 8232} \right).} \hfill \cr }

Using the above information, the characteristic polynomial of the signless Laplacian matrix of the proper power graph of M₁₁ is given as follows: $\begin{array}{l} Θ (Q (𝒫 (M_{11})), x) & = Θ (Q (𝒫 (ℤ_{2})), x) \times Θ (Q (𝒫 (ℤ_{4})), x) \times Θ (Q (𝒫 (ℤ_{10})), x) \times Θ (Q (L^{*}), x) \\ = {(x (x - 2))}^{55} \times {((x - 6) {(x - 2)}^{3})}^{396} \\ \times {({(x - 8)}^{5} {(x - 7)}^{3} (x^{2} - 21 x + 70))}^{144} \\ \times ({(x - 3)}^{4} {(x - 5)}^{18} (x - 8) {(x - 11)}^{2} \\ {\times (x^{4} - 52 x^{3} + 849 x^{2} - 5110 x + 8232))}^{165} \\ = x^{55} {(x - 2)}^{1243} {(x - 3)}^{660} {(x - 5)}^{2970} {(x - 6)}^{396} {(x - 7)}^{432} \\ \times {(x - 8)}^{785} {(x - 11)}^{330} {((x^{2} - 21 x + 70))}^{144} \\ \times {(x^{4} - 52 x^{3} + 849 x^{2} - 5110 x + 8232)}^{165} . \end{array}$ \matrix{ {\Theta \left( {Q\left( {{\cal P}\left( {{M_{11}}} \right)} \right),x} \right)} \hfill & { = \Theta \left( {Q\left( {{\cal P}\left( {{_2}} \right)} \right),x} \right) \times \Theta \left( {Q\left( {{\cal P}\left( {{_4}} \right)} \right),x} \right) \times \Theta \left( {Q\left( {{\cal P}\left( {{_{10}}} \right)} \right),x} \right) \times \Theta \left( {Q\left( {{L^*}} \right),x} \right)} \hfill \cr {} \hfill & { = {{\left( {x(x - 2)} \right)}^{55}} \times {{\left( {(x - 6){{(x - 2)}^3}} \right)}^{396}}} \hfill \cr {} \hfill & { \times {{\left( {{{(x - 8)}^5}{{(x - 7)}^3}\left( {{x^2} - 21x + 70} \right)} \right)}^{144}}} \hfill \cr {} \hfill & { \times \left( {{{(x - 3)}^4}{{(x - 5)}^{18}}(x - 8){{(x - 11)}^2}} \right.} \hfill \cr {} \hfill & {{{\left. { \times ({x^4} - 52{x^3} + 849{x^2} - 5110x + 8232)} \right)}^{165}}} \hfill \cr {} \hfill & { = {x^{55}}{{(x - 2)}^{1243}}{{(x - 3)}^{660}}{{(x - 5)}^{2970}}{{(x - 6)}^{396}}{{(x - 7)}^{432}}} \hfill \cr {} \hfill & { \times {{(x - 8)}^{785}}{{(x - 11)}^{330}}{{\left( {({x^2} - 21x + 70)} \right)}^{144}}} \hfill \cr {} \hfill & { \times {{({x^4} - 52{x^3} + 849{x^2} - 5110x + 8232)}^{165}}.} \hfill \cr }

Thus, the signless Laplacian spectrum of the proper power graph of M₁₁ consists of 0 having multiplicity 55, 2 having multiplicity 1243, 3 having multiplicity 660, 5 having multiplicity 2970, 6 having multiplicity 396, 7 having multiplicity 432, 8 having multiplicity 785, 11 having multiplicity 330, α₁, α₂ each having multiplicity 144, and β₁, β₂, β₃, β₄ each having multiplicity 165, where α₁, α₂ are roots of the polynomial equation x² – 21x + 70 = 0, and β₁, β₂, β₃, β₄ are roots of the polynomial equation x⁴ – 52x³ + 849x² – 5110x + 8232 = 0.

Theorem 3.

The signless Laplacian spectrum of 𝒫(Q_2^k+2) is given as follows: $σ (𝒫 (Q_{2^{k + 2}})) = (\begin{matrix} 2 n - 2 & 4 n - 2 & 4 & 2 & α_{1} & α_{2} \\ 2 n - 3 & 1 & n - 1 & n & 1 & 1 \end{matrix}),$ \sigma \left( {{\cal P}\left( {{Q_{{2^{k + 2}}}}} \right)} \right) = \left( {\matrix{ {2n - 2} & {4n - 2} & 4 & 2 & {{\alpha _1}} & {{\alpha _2}} \cr {2n - 3} & 1 & {n - 1} & n & 1 & 1 \cr } } \right), where α₁, α₂ are roots of the cubic polynomial x³ – 8nx² + (16n² + 8n – 12)x – 48n² + 64n – 16.

Proof.

Using [30, Section 4.1], we find that 𝒫(Q_2^k+2) is constructed from a copy of 𝒫(ℤ_2n), and n copies of the complete graph K₄, which share the identity element e of Q_2^k+2, and the element aⁿ known as the involution. Since n = 2^k, hence using [2, Theorem 2.12], we find that 𝒫(ℤ_2n) is the complete graph on 2n vertices. Thus, we have $𝒫 (Q_{2^{k + 2}}) = P_{3} [K_{2 n - 2}, K_{2}, n K_{2}] .$ {\cal P}({Q_{{2^{k + 2}}}}) = {P_3}[{K_{2n - 2}},{K_2},n{K_2}].

Since the characteristic polynomial of C_P3 is x³ – 8nx² + (16n² + 8n – 12)x – 48n² + 64n – 16, the result follows. We now find the signless Laplacian spectrum of power graph of quaternion group.

Corollary 4.

The signless Laplacian spectrum of 𝒫(Q₈) is given as follows: $σ ((Q_{8})) = (\begin{matrix} 4 & 6 & 2 & 10 \\ 3 & 1 & 3 & 1 \end{matrix}) .$ \sigma \left( {{\cal P}\left( {{Q_8}} \right)} \right) = \left( {\matrix{ 4 & 6 & 2 & {10} \cr 3 & 1 & 3 & 1 \cr } } \right).

Proof.

On putting n = 2 in 3, we find that the signless Laplacian spectrum of 𝒫(Q₈) consists of 4 having multiplicity 2, 6 having multiplicity 1, 2 having multiplicity 2, and remaining eigenvalues are roots of the cubic polynomial x³ – 16x² + 68x – 80. Since x³ – 16x² + 68x – 80 = (x – 2)(x – 4)(x – 10), the result follows.

3.2

Bounds on signless Laplacian spectral radius of power graph of dicyclic group

In the following theorem, we shall provide an upper bound and a lower bound on the Q-spectral radius of 𝒫(Dic_n).

Theorem 5.

If n ≥ 3 is a positive integer, then the Q-spectral radius of 𝒫(Dic_n) is bounded above by $\frac{3 n - 6 + \sqrt{8 α + n^{2} - 4 n + 4}}{2} + 2 n + \sqrt{2 n}$ {{3n - 6 + \sqrt {8\alpha + {n^2} - 4n + 4} } \over 2} + 2n + \sqrt {2n} . Moreover, the Q-spectral radius of 𝒫(Dic_n) is bounded below by $\frac{2 α + n - 2 + \sqrt{n^{2} - 4 α^{2} + 4 α n - 4 n + 4}}{2}$ {{2\alpha + n - 2 + \sqrt {{n^2} - 4{\alpha ^2} + 4\alpha n - 4n + 4} } \over 2}. Here, α = ϕ(n) + 1.

Proof.

We note that 𝒫(Dic_4n) consists of one copy of 𝒫(ℤ_2n), and n copies of the complete graph K₄, which contain the elements e and aⁿ of Dic_4n.

We shall index the elements of 𝒫(Dic_4n) in the following order:

We first list the elements e and aⁿ. We then list the remaining elements of ℤ_n namely {a,a²,a³, ⋯,aⁿ⁻¹,aⁿ⁺¹, aⁿ⁺², ⋯,a²ⁿ⁻¹}.

We then list the remaining elements of Dic_4n in the following order: ${b, a^{n} b, a b, a^{n + 1} b, a^{2} b, a^{n + 2} b, \dots, a^{n - 1} b, a^{2 n - 1} b} .$ \left\{ {b,{a^n}b,ab,{a^{n + 1}}b,{a^2}b,{a^{n + 2}}b, \cdots ,{a^{n - 1}}b,{a^{2n - 1}}b} \right\}.

Using the above indexing, the signless Laplacian matrix of 𝒫(Dic_4n) is of the following form: $Q (𝒫 ({Dic}_{4 n})) = [\frac{Q {𝒫 ((ℤ_{n}))}_{2 n \times 2 n} + ℳ}{𝒩_{2 n \times 2 n}^{T}} | \frac{𝒩_{2 n \times 2 n}}{𝒦_{2 n \times 2 n}}],$ Q\left( {{\cal P}\left( {{{{\mathop{\rm Dic}\nolimits} }_{4n}}} \right)} \right) = \left[ {{{Q{{\left( {{\cal P}\left( {{_n}} \right)} \right)}_{2n \times 2n}} + {\cal M}} \over {{\cal N}_{2n \times 2n}^T}}\left| {{{{{\cal N}_{2n \times 2n}}} \over {{{\cal K}_{2n \times 2n}}}}} \right.} \right], where $ℳ = [\begin{matrix} 2 n & 0 & 0 & 0 & \dots & \dots & 0 \\ 0 & 2 n & 0 & 0 & \dots & \dots & 0 \\ 0 & 0 & 0 & 0 & \dots & \dots & 0 \\ 0 & 0 & 0 & 0 & \dots & \dots & 0 \\ \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}], 𝒩 = [\begin{matrix} 1 & 1 & 1 & 1 & \dots & \dots & 1 \\ 1 & 1 & 1 & 1 & \dots & \dots & 1 \\ 0 & 0 & 0 & 0 & \dots & \dots & 0 \\ 0 & 0 & 0 & 0 & \dots & \dots & 0 \\ \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]$ {\cal M} = \left[ {\matrix{ {2n} & 0 & 0 & 0 & \cdots & \cdots & 0 \cr 0 & {2n} & 0 & 0 & \cdots & \cdots & 0 \cr 0 & 0 & 0 & 0 & \cdots & \cdots & 0 \cr 0 & 0 & 0 & 0 & \cdots & \cdots & 0 \cr \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \cr \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr } } \right],{\cal N} = \left[ {\matrix{ 1 & 1 & 1 & 1 & \cdots & \cdots & 1 \cr 1 & 1 & 1 & 1 & \cdots & \cdots & 1 \cr 0 & 0 & 0 & 0 & \cdots & \cdots & 0 \cr 0 & 0 & 0 & 0 & \cdots & \cdots & 0 \cr \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \cr \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr } } \right] and $𝒦 = [\begin{matrix} 3 & 1 & 0 & 0 & \dots & \dots & \dots & \dots & 0 \\ 1 & 3 & 0 & 0 & \dots & \dots & \dots & \dots & 0 \\ 0 & 0 & 3 & 1 & \dots & \dots & \dots & \dots & 0 \\ 0 & 0 & 1 & 3 & \dots & \dots & \dots & \dots & 0 \\ \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ 0 & 0 & 0 & 0 & \dots & \dots & \dots & 3 & 1 \\ 0 & 0 & 0 & 0 & \dots & \dots & \dots & 1 & 3 \end{matrix}] .$ {\cal K} = \left[ {\matrix{ 3 & 1 & 0 & 0 & \ldots & \ldots & \ldots & \ldots & 0 \cr 1 & 3 & 0 & 0 & \ldots & \ldots & \ldots & \ldots & 0 \cr 0 & 0 & 3 & 1 & \ldots & \ldots & \ldots & \ldots & 0 \cr 0 & 0 & 1 & 3 & \ldots & \ldots & \ldots & \ldots & 0 \cr \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \cr \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \cr \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \cr \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \cr 0 & 0 & 0 & 0 & \ldots & \ldots & \ldots & 3 & 1 \cr 0 & 0 & 0 & 0 & \ldots & \ldots & \ldots & 1 & 3 \cr } } \right].

Using [31, Theorem 2.8.1] and $σ (𝒦) = (\begin{array}{l} 2 & 4 \\ n & n \end{array})$ \sigma ({\cal K}) = \left( {\matrix{ 2 \hfill & 4 \hfill \cr n \hfill & n \hfill \cr } } \right), we obtain 6 $\begin{array}{l} λ_{n} (Q (𝒫 ({Dic}_{n}))) & \leq λ_{n} ([\begin{matrix} Q (𝒫 (ℤ_{n})) + ℳ & 0 \\ 0 & 𝒦 \end{matrix}]) + λ_{n} ([\begin{matrix} 0 & 𝒩 \\ 𝒩^{T} & 0 \end{matrix}]) \\ = \max {λ_{n} (Q (𝒫 (ℤ_{n})) + ℳ), 4} + λ_{n} ([\begin{matrix} 0 & 𝒩 \\ 𝒩^{T} & 0 \end{matrix}]) \\ \leq λ_{n} (Q (𝒫 (ℤ_{n})) + λ_{n} (ℳ) + λ_{n} ([\begin{matrix} 0 & V \\ V^{T} & (4 n - 5) I \end{matrix}]) \\ \leq λ_{n} (Q (𝒫 (ℤ_{n})) + 2 n + λ_{n} ([\begin{matrix} 0 & 𝒩 \\ 𝒩^{T} & 0 \end{matrix}]) . \end{array}$ \matrix{ {{\lambda _n}\left( {Q\left( {{\cal P}\left( {{{{\mathop{\rm Dic}\nolimits} }_n}} \right)} \right)} \right)} \hfill & { \le {\lambda _n}\left( {\left[ {\matrix{ {Q\left( {{\cal P}\left( {{_n}} \right)} \right) + {\cal M}} & 0 \cr 0 & {\cal K} \cr } } \right]} \right) + {\lambda _n}\left( {\left[ {\matrix{ 0 & {\cal N} \cr {{{\cal N}^T}} & 0 \cr } } \right]} \right)} \hfill \cr {} \hfill & { = \max \left\{ {{\lambda _n}\left( {Q\left( {{\cal P}\left( {{_n}} \right)} \right) + {\cal M}} \right),4} \right\} + {\lambda _n}\left( {\left[ {\matrix{ 0 & {\cal N} \cr {{{\cal N}^T}} & 0 \cr } } \right]} \right)} \hfill \cr {} \hfill & { \le {\lambda _n}\left( {Q\left( {{\cal P}\left( {{_n}} \right)} \right) + {\lambda _n}({\cal M}) + {\lambda _n}\left( {\left[ {\matrix{ 0 & V \cr {{V^T}} & {(4n - 5)I} \cr } } \right]} \right)} \right.} \hfill \cr {} \hfill & { \le {\lambda _n}\left( {Q\left( {{\cal P}\left( {{_n}} \right)} \right) + 2n + {\lambda _n}\left( {\left[ {\matrix{ 0 & {\cal N} \cr {{{\cal N}^T}} & 0 \cr } } \right]} \right).} \right.} \hfill \cr }

Using Schur complement determinant formula [31, Theorem 2.7.1], the matrix $[\begin{matrix} 0 & 𝒩 \\ 𝒩^{T} & 0 \end{matrix}]$ \left[ {\matrix{ 0 & {\cal N} \cr {{{\cal N}^T}} & 0 \cr } } \right] has characteristic polynomial as: 7 $\begin{array}{l} Θ (x) & = \det ([\begin{matrix} x I & - 𝒩 \\ - 𝒩^{T} & x I \end{matrix}]) \\ = \det (x I) \times \det (x I - \frac{𝒩 𝒩^{T}}{x}) \\ = \det (x I) \times \det (x I - \frac{1}{x} (\begin{matrix} n & n & 0 & 0 & \dots & \dots & 0 \\ n & n & 0 & 0 & \dots & \dots & 0 \\ 0 & 0 & 0 & 0 & \dots & \dots & 0 \\ 0 & 0 & 0 & 0 & \dots & \dots & 0 \\ 0 & 0 & 0 & 0 & \dots & \dots & 0 \\ 0 & 0 & 0 & 0 & \dots & \dots & 0 \\ 0 & 0 & 0 & 0 & \dots & \dots & 0 \end{matrix})) \\ = \det (x I) \times \det (\begin{matrix} x - \frac{n}{x} & - \frac{n}{x} & 0 & 0 & \dots & \dots & 0 \\ - \frac{n}{x} & x - \frac{n}{x} & 0 & 0 & \dots & \dots & 0 \\ 0 & 0 & x & 0 & \dots & \dots & 0 \\ 0 & 0 & 0 & x & \dots & \dots & 0 \\ 0 & 0 & 0 & 0 & \dots & \dots & 0 \\ 0 & 0 & 0 & 0 & \dots & \dots & 0 \\ 0 & 0 & 0 & 0 & \dots & \dots & x \end{matrix}) \\ = x^{n} \times x^{n - 2} \times ({(x - \frac{n}{x})}^{2} - {(\frac{n}{x})}^{2}) \\ = x^{2 n - 2} (x^{2} - 2 n) . \end{array}$ \matrix{ {\Theta (x)} \hfill & { = \det \left( {\left[ {\matrix{ {xI} & { - {\cal N}} \cr { - {{\cal N}^T}} & {xI} \cr } } \right]} \right)} \hfill \cr {} \hfill & { = \det (xI) \times \det \left( {xI - {{{\cal N}{{\cal N}^T}} \over x}} \right)} \hfill \cr {} \hfill & { = \det (xI) \times \det \left( {xI - {1 \over x}\left( {\matrix{ n & n & 0 & 0 & \cdots & \cdots & 0 \cr n & n & 0 & 0 & \cdots & \cdots & 0 \cr 0 & 0 & 0 & 0 & \cdots & \cdots & 0 \cr 0 & 0 & 0 & 0 & \cdots & \cdots & 0 \cr 0 & 0 & 0 & 0 & \cdots & \cdots & 0 \cr 0 & 0 & 0 & 0 & \cdots & \cdots & 0 \cr 0 & 0 & 0 & 0 & \cdots & \cdots & 0 \cr } } \right)} \right)} \hfill \cr {} \hfill & { = \det (xI) \times \det \left( {\matrix{ {x - {n \over x}} & { - {n \over x}} & 0 & 0 & \cdots & \cdots & 0 \cr { - {n \over x}} & {x - {n \over x}} & 0 & 0 & \cdots & \cdots & 0 \cr 0 & 0 & x & 0 & \cdots & \cdots & 0 \cr 0 & 0 & 0 & x & \cdots & \cdots & 0 \cr 0 & 0 & 0 & 0 & \cdots & \cdots & 0 \cr 0 & 0 & 0 & 0 & \cdots & \cdots & 0 \cr 0 & 0 & 0 & 0 & \cdots & \cdots & x \cr } } \right)} \hfill \cr {} \hfill & { = {x^n} \times {x^{n - 2}} \times \left( {{{\left( {x - {n \over x}} \right)}^2} - {{\left( {{n \over x}} \right)}^2}} \right)} \hfill \cr {} \hfill & { = {x^{2n - 2}}\left( {{x^2} - 2n} \right).} \hfill \cr }

Using 7, we find that the spectrum of $[\begin{matrix} 0 & 𝒩 \\ 𝒩^{T} & 0 \end{matrix}]$ \left[ {\matrix{ 0 & {\cal N} \cr {{{\cal N}^T}} & 0 \cr } } \right] consists of 0 having multiplicity 2n – 2 and $\pm \sqrt{2 n}$ \sqrt {2n} having multiplicity 1. Hence, the largest eigenvalue of $[\begin{matrix} 0 & 𝒩 \\ 𝒩^{T} & 0 \end{matrix}]$ \left[ {\matrix{ 0 & {\cal N} \cr {{{\cal N}^T}} & 0 \cr } } \right] is $\sqrt{2 n}$ \sqrt {2n} . Therefore, using 6 and [32, Theorem 2.1], we obtain $\begin{array}{l} λ_{n} (Q (𝒫 ({Dic}_{n}))) & \leq λ_{n} (Q (𝒫 (ℤ_{n})) + 2 n + \sqrt{2 n} \\ \leq \frac{3 n - 6 + \sqrt{8 α + n^{2} - 4 n + 4}}{2} + 2 n + \sqrt{2 n} . \end{array}$ \matrix{ {{\lambda _n}\left( {Q\left( {{\cal P}\left( {{{{\mathop{\rm Dic}\nolimits} }_n}} \right)} \right)} \right)} \hfill & { \le {\lambda _n}\left( {Q\left( {{\cal P}\left( {{_n}} \right)} \right) + 2n + \sqrt {2n} } \right.} \hfill \cr {} \hfill & { \le {{3n - 6 + \sqrt {8\alpha + {n^2} - 4n + 4} } \over 2} + 2n + \sqrt {2n} .} \hfill \cr }

It is quite trivial to note that the number of edges in 𝒫(Dic_n) is greater than that in 𝒫(ℤ_n). Hence, using Interlacing Theorem [33, Theorem 7.8.13], we find that $λ_{n} (Q 𝒫 ((ℤ_{n}))) \leq λ_{n} (Q 𝒫 (({Dic}_{n}))) .$ {\lambda _n}\left( {Q\left( {{\cal P}\left( {{_n}} \right)} \right)} \right) \le {\lambda _n}\left( {Q\left( {{\cal P}\left( {{{{\mathop{\rm Dic}\nolimits} }_n}} \right)} \right)} \right).

Moreover, using [32, Theorem 2.2], $λ_{n} (Q (𝒫 (ℤ_{n}))) \geq \frac{2 α + n - 2 + \sqrt{n^{2} - 4 α^{2} + 4 α n - 4 n + 4}}{2} .$ {\lambda _n}(Q(({_n}))) \ge {{2\alpha + n - 2 + \sqrt {{n^2} - 4{\alpha ^2} + 4\alpha n - 4n + 4} } \over 2}.

Combining the two equations, we obtain, $λ_{n} (Q (({Dic}_{n}))) \geq \frac{2 α + n - 2 + \sqrt{n^{2} - 4 α^{2} + 4 α n - 4 n + 4}}{2} .$ {\lambda _n}\left( {Q\left( {{\cal P}\left( {{{{\mathop{\rm Dic}\nolimits} }_n}} \right)} \right)} \right) \ge {{2\alpha + n - 2 + \sqrt {{n^2} - 4{\alpha ^2} + 4\alpha n - 4n + 4} } \over 2}.

Hence, the result follows.

4

Conclusion and future work

The universal adjacency matrix of a graph 𝒢 denoted by 𝒰(𝒢) is defined as 𝒰(𝒢) = αA (𝒢) + βI + γJ + δDeg(𝒢), where α, β, γ, δ ∈ ℝ and α ≠ 0, I is the identity matrix and J is the matrix all of whose entries are 1. It was introduced by Haemers and Omidi in [34] to unify the approach to study the spectral theories of the adjacency, Laplacian, and signless Laplacian spectra of graphs. Interested readers may determine the universal adjacency spectrum of power graph of various finite groups considered in this paper. Moreover, determining energy of graphs associated with algebraic structures has also attracted the attention of researchers over the last few decades, see for example [35] and [36].

The signless Laplacian spectrum of power graph of different finite non-commutative groups was examined in this research article. First, for elementary abelian groups whose orders are powers of a prime integer, we found the spectrum of the signless Laplacian matrix of the power graph. Next, we determined the signless Laplacian spectrum of the Mathieu group M₁₁, which is the smallest sporadic group. Additionally, we got the signless Laplacian eigenvalues of 𝒫(Q_2^k+2), where the generalized quaternion group is denoted by Q_2^k+2. At last, we established constraints on the signless Laplacian spectral radius of 𝒫(Dic_4n), where Dic_4n is the dicyclic group.

We encourage the readers to determine the adjacency, Laplacian, and signless Laplacian energy of power graph of various finite groups considered in this paper.

5

Declaration

Signless Laplacian spectrum of power graph of certain finite non-commutative groups

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