Diameter of the commuting graph of $\mathbb{R}^{n\times n}$

TL;DR

This paper determines that the diameter of the commuting graph of real n-by-n matrices (for n≥3) is exactly four, providing a concise proof for this mathematical property.

Contribution

The paper establishes the precise diameter of the commuting graph of real matrices for n≥3, confirming it is exactly four, which was previously only bounded.

Findings

The diameter of the commuting graph is exactly four for n≥3.

Provides a short proof confirming the diameter value.

Clarifies the structure of commuting relations in real matrix algebras.

Abstract

The vertices of commuting graph of $\mathbb{R}^{n\times n}$ are non-scalar matrices; the edges are defined as pairs $(u, v)$ satisfying $uv = vu$. For $n\geqslant 3$, the diameter of this graph is at least four; we give a short proof that it is exactly four.