Can connected commuting graphs of finite groups have arbitrarily large diameter ?

TL;DR

This paper explores the possibility of finite groups with connected commuting graphs having arbitrarily large diameters, challenging a previous conjecture and supported by heuristic reasoning and simulation results.

Contribution

It introduces a family of finite groups potentially with large commuting graph diameters, providing heuristic arguments and simulation data that suggest the conjecture may be false.

Findings

Simulations found groups with commuting graph diameters up to 10

No known finite group previously had a commuting graph diameter greater than 6

Heuristic arguments support the possibility of arbitrarily large diameters

Abstract

We present a family of finite, non-abelian groups and propose that there are members of this family whose commuting graphs are connected and of arbitrarily large diameter. If true, this would disprove a conjecture of Iranmanesh and Jafarzadeh. While unable to prove our claim, we present a heuristic argument in favour of it. We also present the results of simulations which yielded explicit examples of groups whose commuting graphs have all possible diameters up to and including 10. Previously, no finite group whose commuting graph had diameter greater than 6 was known.