Unital compressed commuting graph of $3 \times 3$ matrices over a finite prime field
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| Format: | Preprint |
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2026
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| _version_ | 1866908819198574592 |
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| author | Boroja, Ivan-Vanja Bukovšek, Damjana Kokol Stopar, Nik |
| author_facet | Boroja, Ivan-Vanja Bukovšek, Damjana Kokol Stopar, Nik |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_07467 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Unital compressed commuting graph of $3 \times 3$ matrices over a finite prime field Boroja, Ivan-Vanja Bukovšek, Damjana Kokol Stopar, Nik Rings and Algebras Combinatorics 15A27, 05C50, 51E15 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. |
| title | Unital compressed commuting graph of $3 \times 3$ matrices over a finite prime field |
| topic | Rings and Algebras Combinatorics 15A27, 05C50, 51E15 |
| url | https://arxiv.org/abs/2602.07467 |