The topological index of a graph gives its topological property that remains invariant up to graph automorphism. The topological indices which are based on the eccentricity of a chemical graph are molecular descriptors that remain constant in the whole molecular structure and therefore have a significant position in chemical graph theory. In recent years, various topological indices are intensively studied for a variety of graph structures. In this article, we will consider graph structures associated with zero-divisors of commutative rings, called zero-divisor graphs. We will compute the topological indices for a class of zero-divisor graphs of finite commutative rings that are based on their edge eccentricity. More precisely, we will compute the first and third index of Zagreb eccentricity, the eccentricity index of geometric arithmetic, the atomic bonding connectivity eccentricity index, and the eccentric harmonic index of the fourth type related to graphs constructed using zero-divisors of finite commutative rings ${\mathbb{Z}}_{p^n}.$