Combinatorial properties of power graphs of finite groups of prime order elements

Abstract

For a finite group G, the power graph P(G) represents the graph with elements of G as $V(P(G))$ and $\mathit{uv}\in E(P(G))$ if and only if one of u or v is a power of the other. In this paper, some combinatorial properties of power graph of a group G having all nonidentity elements of prime order are investigated. The necessary and sufficient conditions for P(G) to be power graph of such group are obtained. Further, unicylic, bicyclic and cacti power graphs of finite groups are characterized. Moreover, the eigenspectra of power graphs of these groups are also computed.

Keywords:

• Circumference
• clique
• elementary abelian p-groups
• maximal subgraphs
• nilpotent groups
• power graph
• Sylow p-subgroups

MSC2010:

• 15A18
• 05C50
• 15A45
• 05C35

Acknowledgments

The authors are thankful to the reviewers for their useful comments, corrections and suggestions, which resulted in an improved version of this paper.