Automorphism of Zero-Divisor Graphs of Nilradicals of Semilocal Ring

Abstract

A zero-divisor graph of a commutative ring R denoted as Γ(R), is a graph whose vertices are the zero divisors of the ring. Any two distinct vertices of the graph are incident if and only if their product is zero. Zero-divisor graphs provide a powerful interface between commutative algebra and graph theory by encoding algebraic annihilation relations into combinatorial structures. While the graph-theoretic properties of Γ(R) have been extensively studied, comparatively little is known about their automorphism groups, particularly for graphs arising from nilradicals of semilocal rings. In this paper, we investigate the automorphism groups of Γ(R) associated with the nonzero nilradical of the semilocal ring Zpnqn, where p ̸= q are primes and n ≥ 2. We show that the valuation structure induced by the prime-power decomposition yields a canonical partition of the vertex set into invariant layers. This stratification rigidly constrains graph automorphisms and forces the automorphism group to decompose as a direct product of symmetric groups indexed by valuation levels. Explicit formulas for these automorphism groups are obtained, thereby extending and unifying earlier results for local rings.