Connectivity and Distance Properties of Zero-Divisor Graphs of Finite Commutative Semilocal Rings

Abstract

  The zero-divisor graph of a commutative ring provides a natural connection between algebraic and graph-theoretic structures. Although extensive research has been conducted on the algebraic and combinatorial properties of zero-divisor graphs, their connectivity and metric properties over finite semilocal rings remain comparatively less explored. In this paper, we investigate the zero-divisor graph associated with the finite semilocal ring \( R=\mathbb Z_{p^{n}q^{m}}, \) where \(p\) and \(q\) are distinct primes and \(n,m\ge2\). Using a valuation-theoretic approach, we establish a natural decomposition of the vertex set into valuation layers. This decomposition enables us to determine the connectivity, shortest-path structure, eccentricities of valuation layers, diameter, radius, centre, and peripheral vertices of the graph. The results demonstrate that the valuation-layer decomposition completely governs the structural, metric, and symmetry properties of the zero-divisor graph, providing a unified framework for the study of finite semilocal rings and extending several known results on zero-divisor graphs.