Non-Isomorphic Groups with Isomorphic Power and Commuting Graphs

Abstract

The power graph of a group $G$ is a graph with vertex set $G$, in which two vertices are adjacent if one is some power of the other. In the commuting graph, with $G$ as the vertex set, two vertices are joined by an edge if they commute in $G$. The enhanced power graph of a group $G$ is a graph with vertex set $G$ and an edge joining two vertices $x$ and $y$ if $\langle x,y\rangle$ is cyclic. In this paper, we answer a question posed by P. J. Cameron, namely, if there exist groups $G$ and $H$ such that the power graph of $G$ is isomorphic to the commuting graph of $H$. We show that the answer is yes if $G$ is the generalised quaternion group and $H$ is the dihedral group. We also show that the enhanced power graph of the dicyclic group is isomorphic to the commuting graph of the dihedral group.