On the diameter of the compressed zero-divisor graph

Abstract

ABSTRACTLet R be a commutative ring with nonzero identity. The relation on R given by a∼b if and only if is an equivalence relation. The compressed zero-divisor graph ΓE(R) of R is the (undirected) graph with vertices the equivalence classes induced by ∼ other than [0]R and [1]R, and distinct vertices [a]R and [b]R are adjacent if and only if ab = 0. The distance between vertices [a]R and [b]R (not necessarily distinct from a) is the length of the shortest path connecting them, and the diameter of the graph, diam(ΓE(R)), is the sup of these distances. In this paper, we continue study of the diameter of the compressed zero-divisor graph ΓE(R). A complete characterization for the possible diameters of ΓE(R) is given exclusively in terms of the ideals of R. Also we give a complete characterization for the possible diameters of ΓE(R[x]) in terms of the diameters of ΓE(R). For a reduced ring R with nonzero zero-divisors, it is shown that .