The Steiner antipodal number of zero-divisor graphs of finite commutative rings

Let $R$ be a finite commutative ring with unity. The zero-divisor graph $\mathrm{\Gamma }(R)$ is defined such that its vertex set comprises all nonzero zero-divisors of $R$, with two distinct vertices being adjacent if and only if their product is zero. This study provides a closed-form expression for the $n$-eccentricity of each vertex and computes the Steiner antipodal number of $\mathrm{\Gamma }(R)$ under the following conditions: (ⅰ) $R={\mathbb{Z}}_m$, (ⅱ) $R$ is a reduced ring, and (ⅲ) $R$ is a finite direct product of rings of the form ${\mathbb{Z}}_m$. Moreover, we establish the existence of a zero-divisor graph with a Steiner antipodal number equal to some positive integer $m$.