Some remarks on the compressed zero-divisor graph

Let R   be a commutative ring with 1≠01≠0. The zero-divisor graph Γ(R)Γ(R) of R is the (undirected) graph with vertices the nonzero zero-divisors of R, and distinct vertices r and s   are adjacent if and only if rs=0rs=0. The relation on R   given by r∼sr∼s if and only if annR(r)=annR(s)annR(r)=annR(s) is an equivalence relation. The compressed zero-divisor graph ΓE(R)ΓE(R) of R   is the (undirected) graph with vertices the equivalence classes induced by ∼ other than [0] and [1], and distinct vertices [r][r] and [s][s] are adjacent if and only if rs=0rs=0. Let RERE be the set of equivalence classes for ∼ on R  . Then RERE is a commutative monoid with multiplication [r][s]=[rs][r][s]=[rs]. In this paper, we continue our study of the monoid RERE and the compressed zero-divisor graph ΓE(R)ΓE(R). We consider several equivalence relations on R   and their corresponding graph-theoretic translations to Γ(R)Γ(R). We also show that the girth of ΓE(R)ΓE(R) is three if it contains a cycle and determine the structure of ΓE(R)ΓE(R) when it is acyclic and the monoids RERE when ΓE(R)ΓE(R) is a star graph.