On the Nonorientable Genus of the Generalized Unit and Unitary Cayley Graphs of a Commutative Ring

Let [Formula: see text] be a commutative ring and the multiplicative group of unit elements text]. In 2012, Khashyarmanesh et al. defined generalized unitary Cayley graph, text], corresponding to subgroup nonempty subset with as graph vertex set text]and two distinct vertices being adjacent if only there exists such that this paper, we characterize all Artinian rings for which is projective. This leads us determine whose graphs, graphs co-maximal are addition, prove an has finite nonorientable genus, must ring. Finally, it proved given positive integer number genus finite.