Recognizing the Commuting Graph of a Finite Group

TL;DR

This paper investigates the problem of determining whether a given graph can be realized as the commuting graph of a finite group, providing efficient algorithms for specific cases like extraspecial groups.

Contribution

It introduces a polynomial-time algorithm for recognizing commuting graphs of extraspecial groups and discusses general observations on the realizability problem.

Findings

Polynomial-time algorithm for extraspecial groups

Insights into the structure of commuting graphs

General observations on the realizability problem

Abstract

In this paper we study the realizability question for commuting graphs of finite groups: Given an undirected graph $X$ is it the commuting graph of a group $G$? And if so, to determine such a group. We seek efficient algorithms for this problem. We make some general observations on this problem, and obtain a polynomial-time algorithm for the case of extraspecial groups.