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

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.

💡 Research Summary

The paper provides a complete description of the unital compressed commuting graph Λ₁ of the matrix ring M₃(GF(p)), where p is a prime. The authors begin by recalling the definition of the ordinary commuting graph Γ(R) for a unital ring R and the notion of the unital compressed commuting graph introduced in earlier work. In Λ₁, vertices correspond to equivalence classes of elements that generate the same unital subring; two vertices are adjacent if any representatives commute.

A key observation is that two matrices generate the same unital subring if and only if they share the same Jordan canonical form over GF(p). Consequently, the vertex set splits into eight types (labelled A–H) according to the factorisation of the characteristic and minimal polynomials. For each type the authors compute:

• the dimension of the generated subring ⟨A⟩,
• the number of generators inside ⟨A⟩,
• the centralizer C(A), and
• the size of the GL₃‑orbit Oₐ.

Using the formula |V_X| = |Oₐ|·ωₐ·|GL₃| / (|C(A)|·|GL₃|)·ωₐ⁻¹ (where ωₐ = |⟨A⟩∩Oₐ|), they obtain exact counts of vertices of each type. Types B, C, F, G, H correspond to non‑scalar derogatory matrices; for these the centralizer equals the generated subring, so they form a dense core of the graph.

The central contribution lies in interpreting the core subgraph (the vertices of types C, F, G, H) as the incidence graph of the projective plane PG(2, p). A bijection is established between each vertex and a point‑line pair (P, ℓ) in the plane: two vertices are adjacent precisely when the corresponding pairs share the same line. This geometric model yields immediate structural information: the core has p² + p + 1 vertices, each of degree p + 1, diameter 2, and is highly symmetric under the action of PGL₃(GF(p)).

The remaining vertex types (A, D, E, B) are attached to the core by analysing their centralizers and the way their generated subrings intersect the core’s subrings. Type A consists of scalar matrices and contributes a single isolated vertex with a loop. Types D and E (single Jordan blocks of size three or a size‑two block plus a scalar) attach as star‑like structures, while type B (two distinct eigenvalues, one repeated) connects via a bipartite pattern determined by the eigenvalue multiplicities.

Section 6 presents an explicit algorithm that builds the whole Λ₁ graph: first construct the projective‑plane core, then iteratively insert vertices of the other types, linking them according to the previously derived neighborhood rules. The algorithm runs in polynomial time (O(p³)) and produces a graph that exactly matches the theoretical description.

Finally, the authors show how to recover the ordinary commuting graph Γ(M₃(GF(p))) from Λ₁. Each vertex ⟨A⟩ of Λ₁ expands to the set ⟨A⟩ \ {scalar multiples of the identity}, which forms a clique in Γ. By blowing up every Λ₁ vertex into its corresponding clique, deleting the central scalar vertex and all loops, the full commuting graph is obtained. The resulting Γ has (p³ – p) vertices, (p³ – p)(p² – 1)/2 edges, diameter 3, and maximal cliques of size p³ – p. This resolves a problem that had remained open for several years.

Overall, the paper combines linear algebra, finite field theory, projective geometry, and combinatorial counting to give a complete, constructive description of both the compressed and ordinary commuting graphs for 3×3 matrices over a prime field, and it provides tools that may be adapted to larger matrix sizes or other finite fields.