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

Let R be a commutative ring, U(R) be the set of all unit elements of R, G be a multiplicative subgroup of U(R) and S be a non-empty subset of G such that S (-1)={s (-1): saS}a <<...S. In [16], K. Khashyarmanesh et al. defined a graph of R, denoted by I"(R,G,S), which generalizes both unit and unitary Cayley graphs of R. In this paper, we derive several bounds for the genus of I"(R,U(R),S). Moreover, we characterize all commutative Artinian rings R for which the genus of I"(R,U(R),S) is one. This leads to the characterization of all commutative Artinian rings whose unit and unitary Cayley graphs have genus one.