Non-Isomorphic Groups with Isomorphic Power and Commuting Graphs

TL;DR

This paper constructs examples of non-isomorphic groups whose power, commuting, and enhanced power graphs are isomorphic, answering a question about the relationships between these graph structures in group theory.

Contribution

It demonstrates the existence of non-isomorphic groups with isomorphic power and commuting graphs, specifically involving quaternion, dihedral, and dicyclic groups.

Findings

Power graph of quaternion group is isomorphic to commuting graph of dihedral group.

Enhanced power graph of dicyclic group is isomorphic to commuting graph of dihedral group.

Examples show non-isomorphic groups can have identical associated 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.