On The Genus of Extended Dot Product Graph and The Crosscap of Dot Product Graph

Let S=C×⋯×C (k times) and 1≤k<∞ be an integer, where C is a commutative ring with unity 1≠0. In this paper, we classify the rings S such that the extended total dot product graph TD¯¯¯¯¯¯¯¯(S) and the extended zero-divisor dot product graph ZD¯¯¯¯¯¯¯¯(S) have genus one and two. Furthermore, we characterize S such that the total dot product graph TD(S) and the zero-divisor dot product graph ZD(S) have crosscap one and two.