The Undirected Power Graph of a Finite Group

Abstract

The power graph \({\fancyscript{P}}(G)\) of a group \(G\) is the graph which has a vertex set of the group elements and two elements are adjacent if one is a power of the other. Chakrabarty, Ghosh, and Sen proved the main properties of the undirected power graph of a finite group. The aim of this paper is to generalize some results of their work and presenting some counterexamples for one of the problems raised by these authors. It is also proved that the power graph of a \(p\)-group is \(2\)-connected if and only if the group is a cyclic or generalized quaternion group and if \(G\) is a nilpotent group which is not of prime power order then the power graph \({\fancyscript{P}}(G)\) is \(2\)-connected. We also prove that the number of edges of the power graph of the simple groups is less than or equal to the number of edges in the power graph of the cyclic group of the same order. This partially answers to a question in an earlier paper. Finally, we give a complete classification of groups in which the power graph is a union of complete graphs sharing a common vertex.