Malaysian Journal of Mathematical Sciences, September 2025, Vol. 19, No. 3
On The Genus of Extended Dot Product Graph and The Crosscap of Dot Product Graph
Ali, A., Sarmin, N. H., and Ahmad, B.
Corresponding Email: asma_ali2@rediffmail.com
Received date: 10 September 2024
Accepted date: 13 March 2025
Abstract:
Let $S=C \times \cdots \times C$ ($k$ times) and $1\leq k<\infty $ be an integer, where $C$ is a commutative ring with unity $1\neq 0.$ In this paper, we classify the rings $S$ such that the extended total dot product graph $\overline{TD}(S)$ and the extended zero-divisor dot product graph $\overline{ZD}(S)$ 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.
Keywords: dot product graph; genus; crosscap; reduced ring
