  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, $\Gamma(R, G, S)$, corresponding to a multiplicative
subgroup $G$ of $U(R)$ and a non-empty subset $S$ of $G$ with $S^{-1}=\{s^{-1}
\mid s\in S\}\subseteq S$, as the graph with vertex set $R$ and two distinct
vertices $x$ and $y$ are adjacent if and only if there exists $s\in S$ such
that $x+sy \in G$. In this paper, we characterize all Artinian rings $R$ whose
$\Gamma(R,U(R), S)$ 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 $\Gamma(R, U(R), S)$ 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
$\Gamma(R, U(R), S)$ has nonorientable genus $k$ is finite.
