Clique Numbers in Direct Sum Matrix Commuting Graphs: Structural Analysis and Optimal Bounds
Bibliographic record
Abstract
We investigate the clique numbers and structural properties of commuting graphs associated with direct sum matrix rings over finite commutative rings. For a finite commutative ring L with unity, we study the commuting graph \(\Gamma(M(m \oplus m, L))\) whose vertex set consists of all non-central matrices in \(M(m \oplus m, L)\), where two distinct vertices are adjacent if and only if they commute. Our main contributions establish fundamental lower bounds for the clique number \(\omega\Gamma(M(m \oplus m, L)))\) across various ring structures. We prove that for any finite commutative ring R with unity and positive integer \(m \geq 3\), the clique number satisfies \(\omega(\Gamma(M(m, R))) \geq |R|^{2m} - |R|^2\). For rings isomorphic to \({Z}_{p^r}\) where \(r \geq 3\) is odd, we establish the improved bound \(\omega(\Gamma(M(m, R))) \geq \max\{(p^r)^{2m} - p^{2r}, (p^{r-1})^{m^2-m}(p^{r+1})^{m-1}p^{2r} - p^{2r}\}\). When \(r \geq 2\) is even, the bound becomes \(\omega(\Gamma(M(m, R))) \geq \max\{(p^r)^{2m} - p^{2r}, (p^r)^{m^2-1}p^{2r} - p^{2r}\}\). Our approach combines sophisticated matrix-theoretic techniques with graph-theoretic analysis to construct explicit maximal cliques and derive optimal bounds. The results provide new insights into the intersection of algebraic graph theory and matrix ring theory, with potential applications in coding theory and combinatorial optimization.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.011 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.006 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.007 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".