On maximal distances in a commuting graph

TL;DR

This paper investigates the maximum distances in commuting graphs of matrix algebras over algebraically closed fields, revealing that such distances occur only between nonderogatory matrices and characterizing certain matrices via these distances.

Contribution

It provides new insights into the structure of commuting graphs by identifying when maximal distances occur and characterizing matrices through these distances.

Findings

Maximal distance occurs only between nonderogatory matrices.

Characterization of rank-one and semisimple matrices using commuting graph distances.

Maximal distances are not attained between derogatory matrices.

Abstract

We study maximal distances in the commuting graphs of matrix algebras defined over algebraically closed fields. In particular, we show that the maximal distance can be attained only between two nonderogatory matrices. We also describe rank-one and semisimple matrices using the distances in the commuting graph.