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

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. The zero-divisor graph associated with a commutative ring encodes deep algebraic information in a combinatorial framework. In this paper, we investigate the automorphism groups of zero-divisor graphs arising from the nonzero nilradical of finite local rings of the form \(\mathbb{Z}_{p^k}\) . By exploiting the natural p-adic valuation on nilpotent elements, we obtain a canonical stratification of the vertex set into valuation levels. This structure allows for a precise description of graph automorphisms as products of symmetric groups acting on valuation classes. The results provide a complete characterization of graph symmetries in this local setting and establish a foundational case for the broader theory of automorphisms of zero-divisor graphs over finite rings.