Unital compressed commuting graph of $3 \times 3$ matrices over a finite prime field

TL;DR

This paper fully characterizes the unital compressed commuting graph of 3x3 matrices over finite prime fields, revealing its structure and providing an algorithm for its construction, thus solving a long-standing open problem.

Contribution

It introduces a complete description of the unital compressed commuting graph for 3x3 matrices over GF(p), combining algebraic and geometric methods and solving an open problem.

Findings

Complete description of the graph structure

Algorithm for constructing the graph

Resolution of a long-standing open problem

Abstract

In this paper we completely describe the unital compressed commuting graph of the ring $\mathcal{M}_3(\mathrm{GF}(p))$ of $3 \times 3$ matrices over the finite prime field $\mathrm{GF}(p)$. To achieve this we combine methods from linear algebra, field theory, projective geometry and combinatorics. We first partition the set of vertices into types based on the Jordan form and describe the neighborhood of each vertex. The key part of the graph, i.e., the subgraph that corresponds to non-scalar derogatory matrices, is then determined using a bijective correspondence between its vertices and point-line pairs in the projective plane over $\mathrm{GF}(p)$. At the end we explain how the remaining vertices are attached to the key part. We also give an algorithm to construct the whole graph. As a consequence, we describe the usual commuting graph $\Gamma(\mathcal{M}_3(\mathrm{GF}(p)))$, whose structure was an open problem for several years.