Certain properties of the power graph associated with a finite group

Abstract

There are a variety of ways to associate directed or undirected graphs to a group. It may be interesting to investigate the relations between the structure of these graphs and characterizing certain properties of the group in terms of some properties of the associated graph. The power graph $\mathcal{P}(G)$ of a group $G$ is a simple graph whose vertex-set is $G$ and two vertices $x$ and $y$ in $G$ are adjacent if and only if $y=x^m$ or $x=y^m$ for some positive integer $m$. We also pay attention to the subgraph $\mathcal{P}^\ast(G)$ of $\mathcal{P}(G)$ which is obtained by deleting the vertex 1 (the identity element of $G$). In the present paper, we first investigate some properties of the power graph $\mathcal{P}(G)$ and the subgraph $\mathcal{P}^\ast(G)$. We next prove that many of finite groups such as finite simple groups, symmetric groups and the automorphism groups of sporadic simple groups can be uniquely determined by their power graphs among all finite groups. We have also determined up to isomorphism the structure of any finite group $G$ such that the graph $\mathcal{P}^\ast(G)$ is a strongly regular graph, a bipartite graph, a planar graph or an Eulerian graph. Finally, we obtained some infinite families of finite groups such that the graph $\mathcal{P}^\ast(G)$ containing some cut-edges.