On the genus and crosscap of the total graph of commutative rings with respect to multiplication

Document Type : Research Paper

Authors

1 School of Computational Sciences‎, ‎Faculty of Science and Technology‎, ‎JSPM University‎,‎‎ ‎‎India

2 Department of Mathematics and Science Education‎, ‎Faculty of Education‎,‎ Karamano\u{g}lu Mehmetbey University‎, ‎Karaman, Turkey

3 Department of Mathematics, Aligarh Muslim University, Aligarh

Abstract

‎Let $\mathcal{S}$ be a commutative ring and $Z(\mathcal{S})$ be its zero-divisors set‎.
‎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 $\alpha\beta \in Z(\mathcal{S})$‎.
‎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 $\mathcal{S}$)‎.
‎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.

Keywords

References

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