Automorphism of Zero Divisor Graphs of Nilradicals of Commutative Finite Local Rings.

Abstract

A zero-divisor graph of a commutative ringRdenoted asΓ(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. The zero-divisor graph associatedwith a commutative ring encodes deep algebraic information in a combinatorial framework. In this paper, we investigatethe automorphism groups of zero-divisor graphs arising from the nonzero nilradical of finite local rings of the formZpk.By exploiting the naturalp-adic valuation on nilpotent elements, we obtain a canonical stratification of the vertex set intovaluation levels. This structure allows for a precise description of graph automorphisms as products of symmetric groupsacting on valuation classes. The results provide a complete characterization of graph symmetries in this local setting andestablish a foundational case for the broader theory of automorphisms of zero-divisor graphs over finite rings.