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

Abstract

Let G be a finite group. The main supergraph S(G) is a graph with vertex set G in which two vertices x and y are adjacent if and only if o(x)|o(y) or o(y)|o(x). In an earlier paper, the main properties of this graph was obtained. The aim of this paper is to investigate the Hamiltonianity, Eulerianness and 2-connectedness of this graph.

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

“…Motivated by the work of Kelarev and Quinn, Chakrabarty et al considered undirected power graphs (power graph for short) of semigroups [8]. In recent years, there has been considerable interest to the study of power graphs, but the second graph introduced very recently by the present authors [19,20]. In [19], the authors focused on the relationship between power graph and its main supergraph and some basic properties of this graph are studied.…”

“…In recent years, there has been considerable interest to the study of power graphs, but the second graph introduced very recently by the present authors [19,20]. In [19], the authors focused on the relationship between power graph and its main supergraph and some basic properties of this graph are studied. In [20], the automorphism group of this graph in general are computed, but this paper devotes to the study of graph eigenvalues of main supergraph.…”

On connected components and perfect codes of proper order graphs of finite groups

“…In [22], Ma and Su studied the independence number of an order graph. Hamiltonianity and Eulerianness of S (G) were investigated in [12] and spectrum and L-spectrum of S (G) were investigated in [11]. In [3], Asboei and Salehi studied the well-known Thompson's problem and recognized the projective special linear groups and the projective linear groups by their order graphs.…”

Strong metric dimensions for power graphs of finite groups

“…Recently, Hamzeh and Ashrafi [14] studied some properties of the order supergraph, and in particular, they showed that S(G) = P(G) if and only if G is cyclic. Also, in [15], they investigated Hamiltonianity, Eulerianness and 2-connectedness of this graph.…”