Connectivity of the zero-divisor graph for finite rings

Abstract

We study the vertex-connectivity and edge-connectivity of the zero-divisor graph ${\Gamma }_R$ associated to a finite commutative ring $R$. We show that the edge-connectivity of ${\Gamma }_R$ always coincides with the minimum degree, and that vertex-connectivity also equals the minimum degree when $R$ is nonlocal. When $R$ is local, we provide conditions for the equality of all three parameters to hold, give examples showing that the vertex-connectivity can be much smaller than minimum degree, and prove a general lower bound on the vertex-connectivity.