Chromatic Invariants and Complete Subgraph Structure in Commuting Graphs of Block-Diagonal Matrix Rings

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DOI: https://doi.org/10.28924/2291-8639-24-2026-63
Published: Mar 12, 2026
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Eman Almuhur, Manal Al-Labadi, Wasim Audeh, Mohammad Esmael Samie, Chromatic Invariants and Complete Subgraph Structure in Commuting Graphs of Block-Diagonal Matrix Rings, Int. J. Anal. Appl., 24 (2026), 63.

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Eman Almuhur, Manal Al-Labadi, Wasim Audeh, Mohammad Esmael Samie

Abstract

We investigate chromatic invariants within commuting graphs associated with block-diagonal matrix rings over finite commutative rings. For a finite commutative ring \(R\) with identity, we analyze the commuting graph \(\Gamma(M(m \oplus m, R))\) whose vertex set consists of non-central block-diagonal matrices \(A \oplus B\) with \(A, B \in M(m, R)\), where edges represent commutativity relations. Our main contributions establish quantitative bounds for both chromatic as functions of the base ring cardinality. We prove that the commuting graph \(\Gamma(M(m \oplus m, R))\) contains \(|R|^{2m^2} - |R|^2\) vertices and derive the lower bound \(\omega(\Gamma(M(m \oplus m, R))) \geq |R|^{2m} - |R|^2\) by constructing explicit maximal cliques from diagonal matrices. For rings of the form \(\mathbb{Z}_{p^r}\), we finding the lower bound for the chromating number if \(R\) is a finite commutative ring with unity, then \(\chi(\gamma(\Gamma(M(m \oplus m, R))) \geq 3\) also, investigate the chromating number through algebraic properties involving centralizers and construct novel families of maximal cliques using nilpotent elements in \(\mathbb{Z}_{p^r}\). Our results demonstrate unbounded growth of chromatic numbers with increasing ring cardinality and illuminate deep connections between the algebraic structure of block-diagonal matrix rings and combinatorial properties of their associated commuting graphs. These findings extend classical results on commuting graphs of matrix rings to the block-diagonal setting and provide tools for analyzing commutativity patterns in structured matrix algebras.

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