International Journal of Analysis and Applications | 2026
Authors: Almuhur E.; Al-Labadi M.; Audeh W.; Samie M.E.
DOI: 10.28924/2291-8639-24-2026-63
Journal: International Journal of Analysis and Applications
Year: 2026
Publisher: Etamaths Publishing
Document Type: Article
Open Access: All Open Access; Gold Open Access
Cited by: 0
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 Γ(M(m ⊕ m, R)) whose vertex set consists of non-central block-diagonal matrices A ⊕ B with A, B ∈ 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 Γ(M(m ⊕ m, R)) contains |R|2m2 − |R|2 vertices and derive the lower bound ω(Γ(M(m ⊕ m, R))) ≥ |R|2m − |R|2 by constructing explicit maximal cliques from diagonal matrices. For rings of the form Zpr, we finding the lower bound for the chromating number if R is a finite commutative ring with unity, then χ(γ(Γ(M(m ⊕ m, R))) ≥ 3 also, investigate the chromating number through algebraic properties involving centralizers and construct novel families of maximal cliques using nilpotent elements in Zp 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. © 2026 the author(s).
chromatic number; clique number; commuting graph; direct sum; matrix ring