Von Neumann regular graphs associated with rings

Abstract

Let [Formula: see text] be a ring with nonzero identity. By the Von Neumann regular graph of [Formula: see text], we mean the graph that its vertices are all elements of [Formula: see text] such that there is an edge between vertices [Formula: see text] if and only if [Formula: see text] is a Von Neumann regular element of [Formula: see text], denoted by [Formula: see text]. In this paper, the basic properties of [Formula: see text] are investigated and some characterization results regarding connectedness, diameter, girth and planarity of [Formula: see text] are given.