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

Ivan-Vanja Boroja, Damjana Kokol Bukovšek, Nik Stopar

TL;DR

The authors resolve the structure of the unital compressed commuting graph for $3\times3$ matrices over the finite field $\mathrm{GF}(p)$ by classifying vertices via Jordan form into eight types, identifying a central core of non-scalar derogatory matrices that corresponds to point–line pairs in the projective plane $\mathrm{PG}(2,\mathrm{GF}(p))$. They develop a precise combinatorial model using the incidence matrix $T_p$ and its associated geometry to describe inter-type edges, then present an explicit construction algorithm for $\Lambda^1(\mathcal{M}_3(\mathrm{GF}(p)))$ and a blow-up procedure to recover the ordinary commuting graph $\Gamma(\mathcal{M}_3(\mathrm{GF}(p)))$. The paper also provides complete counts of vertices by type, neighborhood structures, and a detailed description of how to assemble the full graph from the $B$–$E$ core and the attached per-type vertices. This work settles long-standing questions about the structure of $\Gamma(\mathcal{M}_3(\mathrm{GF}(p)))$ by leveraging geometry, centralizers, and unital subring generation, with potential implications for isomorphism problems in matrix rings.

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 $Γ(\mathcal{M}_3(\mathrm{GF}(p)))$, whose structure was an open problem for several years.

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

TL;DR

The authors resolve the structure of the unital compressed commuting graph for matrices over the finite field by classifying vertices via Jordan form into eight types, identifying a central core of non-scalar derogatory matrices that corresponds to point–line pairs in the projective plane . They develop a precise combinatorial model using the incidence matrix and its associated geometry to describe inter-type edges, then present an explicit construction algorithm for and a blow-up procedure to recover the ordinary commuting graph . The paper also provides complete counts of vertices by type, neighborhood structures, and a detailed description of how to assemble the full graph from the – core and the attached per-type vertices. This work settles long-standing questions about the structure of by leveraging geometry, centralizers, and unital subring generation, with potential implications for isomorphism problems in matrix rings.

Abstract

In this paper we completely describe the unital compressed commuting graph of the ring of matrices over the finite prime field . 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 . 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 , whose structure was an open problem for several years.
Paper Structure (10 sections, 7 theorems, 79 equations, 1 figure, 2 tables)

This paper contains 10 sections, 7 theorems, 79 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Vertex set of Λ¹(ℳ₃(GF(p)))
  4. Neighborhoods of vertices of Λ¹(ℳ₃(GF(p)))
  5. Subgraph induced on V(B) U V(E)
  6. Projective plane over GF(p) and its incidence matrix
  7. Bijection between V(B) U V(E) and P × L
  8. Geometrical interpretation of edges
  9. Description of Λ¹(ℳ₃(GF(p)))
  10. Commuting graph of ℳ₃(GF(p))

Key Result

Proposition 2.3

Let $A\in \mathcal{M}_n({\mathbb F})$, where ${\mathbb F}$ is a finite field. Then

Figures (1)

  • Figure 1: Possible $2\times 2$ submatrices of $T_p$ and the edges between their entries.

Theorems & Definitions (15)

  • Definition 2.1: AkGhHaMo04
  • Definition 2.2: BoDoKoSt23
  • Proposition 2.3
  • Proposition 2.4: BoDoKoSt23
  • Proposition 3.1
  • proof
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Definition 3.4
  • ...and 5 more