Invariants and isomorphism theorems for zero-divisor graphs of commutative rings of quotients

Abstract

Given a commutative ring $R$ with $1\neq0$, the zero-divisor graph $\Gamma(R)$ of $R$ is the graph whose vertices are the nonzero zero-divisors of $R$, such that distinct vertices are adjacent if and only if their product in $R$ is $0$. It is well known that the zero-divisor graph of any ring is isomorphic to that of its total quotient ring. This result fails for more general rings of quotients. In this paper, conditions are given for determining whether the zero-divisor graph of a ring of quotients of $R$ is isomorphic to that of $R$. Examples involving zero-divisor graphs of rationally $\aleph_0$-complete commutative rings are studied extensively. Moreover, several graph invariants are studied and applied in this investigation.