A matrix ring with commuting graph of maximal diameter

TL;DR

This paper constructs an example of a matrix ring over a field where the commuting graph achieves the maximum diameter of six, illustrating a key property of algebraic structures related to commuting relations.

Contribution

It provides the first known example of a matrix ring with a commuting graph of maximal diameter, advancing understanding of algebraic graph properties.

Findings

Commuting graph of certain matrix rings can have diameter six.

Maximal diameter of the commuting graph is achievable in matrix rings.

The example applies to specific fields and matrix sizes.

Abstract

The commuting graph of a semigroup is the set of non-central elements; the edges are defined as pairs $(u,v)$ satisfying $uv=vu$. We provide an example of a field $F$ and an integer $n$ such that the commuting graph of $\operatorname{Mat}_n(F)$ has maximal possible diameter, equal to six.