A generalization of commuting graphs

Abstract

Let R be a ring with the identity element 1, α be an endomorphism of R and δ be a left α-derivation. In this paper, we introduce a generalization of a commuting graph, which is denoted by ΓR(α, δ), as a directed graph with vertex set R and, for two distinct vertices x and y, there is an arc from x to y if and only if xy = α(y)x + δ(y). We study some basic properties of ΓR(α, δ). Also, we investigate the planarity and genus of the graph ΓR(α, 0).