Diameter, girth and cut vertices of the graph of equivalence classes of zero-divisors

Abstract

We explore the properties of ${\Gamma }_E\left(R\right)$, the graph of equivalence classes of zero-divisors of a commutative Noetherian ring $R$. We determine the possible combinations of diameter and girth for the zero-divisor graph $\Gamma \left(R\right)$ and the equivalence class graph ${\Gamma }_E\left(R\right)$, and examine properties of cut-vertices of ${\Gamma }_E\left(R\right)$.