University of Guilan Journal of Algebra and Related Topics 2345-3931 14 Special Issue- Dedicated to the memory of Jürgen Herzog (1941-2024). 2026 04 01 On the genus and crosscap of the total graph of commutative rings with respect to multiplication 203 215 8190 10.22124/jart.2024.27463.1667 EN M. Nazim School of Computational Sciences‎, ‎Faculty of Science and Technology‎, ‎JSPM University‎,‎‎ ‎‎India C. Abdioglu Department of Mathematics and Science Education‎, ‎Faculty of Education‎,‎ Karamano\u{g}lu Mehmetbey University‎, ‎Karaman, Turkey N. Rehman Department of Mathematics, Aligarh Muslim University, Aligarh Sh. A. Mir Department of Mathematics, Aligarh Muslim University, Aligarh Journal Article 2024 05 13 ‎Let $\mathcal{S}$ be a commutative ring and $Z(\mathcal{S})$ be its zero-divisors set‎.<br />‎The total graph of $\mathcal{S}$ with respect to multiplication‎, ‎denoted by $T_{Z(\mathcal{S})}(\Gamma(\mathcal{S}))$‎, ‎is an undirected graph with vertex set as the ring elements $\mathcal{S}$ and two distinct vertices $\alpha$ and $\beta$ are adjacent if and only if $\alpha\beta \in Z(\mathcal{S})$‎.<br />‎The graph $T_{Z(\mathcal{S})}(\Gamma(\mathcal{S}^*))$ is a subgraph of $T_{Z(\mathcal{S})}(\Gamma(\mathcal{S}))$ with vertex set $\mathcal{S}^*$ (set of nonzero elements of $\mathcal{S}$)‎.<br />‎In this paper‎, ‎we characterize finite rings $\mathcal{S}$ for which $T_{Z(\mathcal{S})}(\Gamma(\mathcal{S}^*))$ belongs to some well-known families of graphs‎. ‎Further‎, ‎we classify the finite rings $\mathcal{S}$ for which $T_{Z(\mathcal{S})}(\Gamma(\mathcal{S}^*))$ is planar‎, ‎toroidal or double toroidal‎. ‎Finally‎, ‎we analyze the finite rings $\mathcal{S}$ for which the graph $T_{Z(\mathcal{S})}(\Gamma(\mathcal{S}^*))$ has crosscap at most two‎. https://jart.guilan.ac.ir/article_8190_c98ff57cf45563db038093829ef851ed.pdf