On the nonorientable genus of the generalized unit and unitary Cayley graphs of a commutative ring

Abstract

Let R be a commutative ring and let U (R) be multiplicative group of unit elements of R. In 2012, Khashyarmanesh et al. defined generalized unit and unitary Cayley graph, Γ(R, G, S), corresponding to a multiplicative subgroup G of U (R) and a non-empty subset S of G with S −1 = {s −1 | s ∈ S} ⊆ S, as the graph with vertex set R and two distinct vertices x and y are adjacent if and only if there exists s ∈ S such that x+sy ∈ G. In this paper, we characterize all Artinian rings R whose Γ(R, U (R), S) is projective. This leads to determine all Artinian rings whose unit graphs, unitary Cayley garphs and co-maximal graphs are projective. Also, we prove that for an Artinian ring R whose Γ(R, U (R), S) has finite nonorientable genus, R must be a finite ring. Finally, it is proved that for a given positive integer k, the number of finite rings R whose Γ(R, U (R), S) has nonorientable genus k is finite.