Zero divisors of Gorenstein Rings

Let $R$ be a commutative Artinian ring. We consider two graphs associated to $R$, namely the compressed zero-divisor graph ${\mathrm{\Gamma }}_E(R)$ and the associate class graph ${\mathrm{\Gamma }}_A(R)$. Partitioning the vertex set of a zero-divisor graph into its core and its boundary, we count the core vertices that dominate the core. This count is a graph invariant, and we estimate it for $\mathrm{\Gamma }(R)$, ${\mathrm{\Gamma }}_A(R)$ and ${\mathrm{\Gamma }}_E(R)$. We prove that the count for ${\mathrm{\Gamma }}_A(R)$ is bounded below by the count for ${\mathrm{\Gamma }}_E(R)$, and that the lower bound is attained precisely when $R$ is Gorenstein. As a consequence we obtain that $R$ is Gorenstein if and only if ${\mathrm{\Gamma }}_A(R)\cong {\mathrm{\Gamma }}_E(R)$ as graphs, the isomorphism being an arbitrary one and not merely the natural compression map. Using the same counting technique we then answer, for Artinian rings, a question of Anderson and LaGrange by showing that $\mathrm{\Gamma }(R)\cong {\mathrm{\Gamma }}_E(R)$ if and only if $R\cong {\mathbb{Z}}_2^n$ for some $n\ge 2$, or $R\cong {\mathbb{Z}}_4$, or $R\cong {\mathbb{Z}}_2[x]\mathrm{/}(x^2)$.