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, Ξ(R,G,S), corresponding to a multiplicative
subgroup G of U(R) and a non-empty subset S of G with Sβ1={sβ1β£sβS}βS, as the graph with vertex set R and two distinct
vertices x and y are adjacent if and only if there exists sβS such
that x+syβG. In this paper, we characterize all Artinian rings R whose
Ξ(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 Ξ(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
Ξ(R,U(R),S) has nonorientable genus k is finite.Comment: To appear in Algebra Colloquiu