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, Pune-412207, India

2 Department of Mathematics Faculty of Education Karamanoglu Mehmetbey University Karaman - Turkey

3 Department of Mathematics Aligarh Muslim University Aligarh

4 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

Journal of Algebra and Related Topics

Articles in Press, Accepted Manuscript
Available Online from 04 December 2024

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