Certain properties of the power graph associated with a finite group. (Q2874704)

The power graph \(\mathcal P(G)\) of a group \(G\) is a simple graph whose vertex set is \(G\) and two vertices in \(G\) are adjacent if and only if one of them is a power of the other. The subgraph \(\mathcal P^*(G)\) of \(\mathcal P(G)\) is obtained by deleting the vertex 1 (the identity element of \(G\)).NEWLINENEWLINE In this paper, the authors investigate some properties of the graphs \(\mathcal P(G)\) and \(\mathcal P^*(G)\) for a finite group \(G\). Some conditions for the graph \(\mathcal P^*(G)\) to be connected are given. A relationship between \(\mathcal P^*(G)\) and two well known graphs associated with the group \(G\) is given: the prime graph of \(G\) and the commuting graph of \(G\). Note that \textit{P. J. Cameron} [J. Group Theory 13, No. 6, 779-783 (2010; Zbl 1206.20023)] showed that the power graph of a finite group determines its spectrum and \textit{M. Mirzargar} et al. [Filomat 26, No. 6, 1201-1208 (2012; Zbl 1289.05211)] proved that each of the finite groups: a simple group, a symmetric group, a cyclic group, a dihedral group, a generalized quaternion group, is determined by its power graph.NEWLINENEWLINE In the given paper, it is proved that the automorphism groups of sporadic simple groups are also determined by the power graph. Next the authors provide necessary and sufficient conditions for a graph \(\mathcal P^*(G)\) to be a strongly regular graph, a bipartite graph or planar graph. Finally, they obtain some infinite families of finite groups \(G\) for which the graph \(\mathcal P^*(G)\) contains some cut-edges.