Clean graph of a ring

Abstract

Let R be a ring (not necessarily commutative) with identity. The clean graph Cl(R) of a ring R is a graph with vertices in form (e,u), where e is an idempotent and u is a unit of R; and two distinct vertices (e,u) and (f,v) are adjacent if and only if ef = fe = 0 or uv = vu = 1. In this paper, we focus on Cl2(R), the subgraph of Cl(R) induced by the set {(e,u): e is a nonzero idempotent element of R}. It is observed that Cl2(R) has a crucial role in Cl(R). The clique number, the chromatic number, the independence number and the domination number of the clean graph for some classes of rings are determined. Moreover, the connectedness and the diameter of Cl2(R) are studied. © 2021 World Scientific Publishing Company.