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).

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Let R be a commutative ring with non-zero identity. We describe all C 3 -and C 4 -free intersection graph of non-trivial ideals of R as well as Cnfree intersection graph when R is a reduced ring. Also, we shall describe all complete, regular and n-claw-free intersection graphs. Finally, we shall prove that almost all Artin rings R have Hamiltonian intersection graphs. We show that such graphs are indeed pancyclic.