Clique Numbers in Direct Sum Matrix Commuting Graphs: Structural Analysis and Optimal Bounds

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.