Non-Isomorphic Groups with Isomorphic Power and Commuting Graphs

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 x, y 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.