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On the genus and crosscap of the total graph of commutative rings with respect to multiplication | ||
| Journal of Algebra and Related Topics | ||
| دوره 14، S1، تیر 2026، صفحه 203-215 اصل مقاله (187.7 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22124/jart.2024.27463.1667 | ||
| نویسندگان | ||
| M. Nazim* 1؛ C. Abdioglu2؛ N. Rehman3؛ Sh. A. Mir3 | ||
| 1School of Computational Sciences, Faculty of Science and Technology, JSPM University, India | ||
| 2Department of Mathematics and Science Education, Faculty of Education, Karamano\u{g}lu Mehmetbey University, Karaman, Turkey | ||
| 3Department of Mathematics, Aligarh Muslim University, Aligarh | ||
| چکیده | ||
| 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. | ||
| کلیدواژهها | ||
| Crosscap of a graph؛ Genus of a graph؛ Total graph with respect to multiplication؛ Zero-divisor graph | ||
| مراجع | ||
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