On the diameter and girth of a zero-divisor graph

For a commutative ring RR with zero-divisors Z(R)Z(R), the zero-divisor graph of RR is Γ(R)=Z(R)−{0}Γ(R)=Z(R)−{0}, with distinct vertices xx and yy adjacent if and only if xy=0xy=0. In this paper, we characterize when either diam(Γ(R))≤2 or gr(Γ(R))≥4. We then use these results to investigate the diameter and girth for the zero-divisor graphs of polynomial rings, power series rings, and idealizations.