Spectrum and L-spectrum of the power graph and its main supergraph for certain finite groups

Abstract

Let G be a finite group. The power graph P(G) and its main supergraph S(G) are two simple graphs with the same vertex set G. Two elements x, y ∈ G are adjacent in the power graph if and only if one is a power of the other. They are joined in S(G) if and only if o(x)|o(y) or o(y)|o(x). The aim of this paper is to compute the characteristic polynomial of these graph for certain finite groups. As a consequence, the spectrum and Laplacian spectrum of these graphs for dihedral, semi-dihedral, cyclic and dicyclic groups were computed.

Recent developments on the power graph of finite groups – a survey

“…Hamzeh et al [47] generalized some results proved in [63] through some more results on power graphs and they are presented below.…”

“…( Hamzeh et al [47] denoted the set of all cyclic subgroups of finite group G by CðGÞ ¼ fC 1 , C 2 , :::C k g be and D G (renamed for the sake of convenience) be the simple undirected graph with vertex set CðGÞ in which two cyclic subgroups are adjacent if one is contained in other. Let K a i be the complete graph of order a i ¼ /ð C i j jÞ: If K G ¼ fK a i j a i ¼ /ð C i j jÞ, C i 2 CðGÞg, then the power graph PðGÞ is isomorphic to D G -join of K a 1 , K a 2 , :::, K a k , see [39] for details.…”

“…In Theorem 9.12, the Laplacian polynomial of PðZ n Þ was obtained. Hamzeh et al [47] applied the Theorem 9.22 and provided a complete description of the Laplacian spectrum of PðZ n Þ in [47, Corollary 3.18]. Also in [47], the authors determined the Laplacian spectrum of PðZ n Þ when n is a prime power and a product of two distinct primes using Theorem 9.22.…”

Signless and normalized Laplacian spectrums of the power graph and its supergraphs of certain finite groups

“…Laplacian spectrum of power graphs of finite cyclic and dihedral groups computed by Chattopadhyaya and Panigrahi in [3] and in [15], Mehranian et al obtained spectrum of the power graphs of D 2n , T 4n and SD 8n . Also in [6,7] the eigenvalues and Laplacian eigenvalues of S(G) are computed. Throughout this paper we refer to [16] for group theory concepts and for graph theoretical concepts and notations, we refer to [19].…”

Some Remarks on the Order Supergraph of the Power Graph of a Finite Group

“…Some basic properties of this graph are studied in [11]. In [9], the automorphism group of this graph computed in general and in [10] its eigenvalues and Laplacian eigenvalues were computed. Set π e (G) = {a 1 , .…”