On the Diameter of the Commuting Graph of a Full Matrix Ring over a Division Ring

TL;DR

This paper investigates the conditions under which the commuting graph of a matrix ring over a division ring is connected and establishes bounds on its diameter when connected.

Contribution

It provides a criterion for connectedness and bounds the diameter of the commuting graph over matrix rings of division rings.

Findings

Connectedness depends on specific conditions of the division ring.

Diameter of the commuting graph is between four and six when connected.

Provides new insights into the structure of commuting graphs over matrix rings.

Abstract

For a division ring D, finite dimensional over its center F, we give a condiction for the connectedness of the commuting graph of a matrix ring over $D$. Furthermore, we prove that if the commuting graph is connected, then its diameter is between four and six.