Automorphism of Zero-DivisorGraphs of Nilradicals of SemilocalRing

Abstract

A zero-divisor graph of a commutative ringRdenoted as Γ(R), is a graph whose vertices are the zero divisorsof the ring. Any two distinct vertices of the graph are incident if and only if their product is zero. Zero-divisorgraphs provide a powerful interface between commutative algebra and graph theory by encoding algebraicannihilation relations into combinatorial structures. While the graph-theoretic properties of Γ(R) have beenextensively studied, comparatively little is known about their automorphism groups, particularly for graphsarising from nilradicals of semilocal rings. In this paper, we investigate the automorphism groups of Γ(R)associated with the nonzero nilradical of the semilocal ringZpnqn, wherep̸=qare primes andn≥2. We show that the valuation structure induced by the prime-power decomposition yields a canonical partition ofthe vertex set into invariant layers. This stratification rigidly constrains graph automorphisms and forcesthe 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 resultsfor local rings.