A Note on a Conjecture about Commuting Graphs

TL;DR

This paper proves that the maximum diameter of the commuting graph of full matrix rings over real numbers is five, confirming a specific conjecture for this case and advancing understanding of algebraic graph structures.

Contribution

It establishes an upper bound of five for the diameter of the commuting graph over real matrices, confirming a conjecture for this particular field.

Findings

Diameter of the commuting graph is at most five.

Confirmed the conjecture for real matrix rings.

Provides insight into algebraic graph structures over real numbers.

Abstract

We prove that the diameter of the commuting graph of the full ma- trix ring over the real numbers is at most five. This answers, in the affir- mative, a conjecture proposed by Akbari-Mohammadian-radjavi-Raja, for the special case of the field of real numbers.