Studying algebraic structures via graphs and hypergraphs assigned to them can be of interest. Especially, computing the genus of a graph as a topological index leads to a better understanding of the related algebraic structure. In this direction we apply a hypergraph, namely $3$ -zero divisor hypergraph assigned to a commutative ring and study its genus. In this paper, we characterize all finite commutative nonlocal rings $A$ with identity whose $ℋ_{3} (A)$ has genus two. Further, we classify all finite commutative nonlocal rings $A$ whose $ℋ_{3} (A)$ has crosscap two. Moreover, we provide a MATLAB code for calculating $3$ -zero-divisor of $ℤ_{n}$ and the hyperedge of $3$ -zero-divisor hypergraph of $ℤ_{n}$ .

Communicated by A. Leroy

Keywords:

AMSC: 05C10, 05C25, 05C65, 13A99

References

1. S. Akbari, H. R. Maimani and S. Yassemi, When a zero-divisor graph is planar or a complete r-partite graph, J. Algebra 270 (2003) 169–180. Crossref, Web of Science, Google Scholar
2. D. F. Anderson, T. Asir, A. Badawi and T. Tamizh Chelvam, Graphs from Rings, 1st edn. (Springer Nature, Switzerland, 2021). Crossref, Google Scholar
3. D. F. Anderson, A. Frazier, A. Lauve and P. S. Livingston, The zero-divisor graph of a commutative ring, II, in Ideal Theoretic Methods in Commutative Algebra, eds. D. D. Anderson and I. J. Patrick (Dekker, New York, 2001), pp. 61–72. Google Scholar
4. D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999) 434–447. Crossref, Web of Science, Google Scholar
5. D. Archdeacon, Topological Graph theory: A survey, Congr. Numer. 115 (1996) 5–54. Google Scholar
6. M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra (Addison-Wesley Publishing Co, London, 1969). Google Scholar
7. I. Beck, Coloring of commutative rings, J. Algebra 116 (1988) 208–226. Crossref, Web of Science, Google Scholar
8. L. W. Beineke and R. J. Wilson, Topics in Topological Graph Theory, Encyclopedia of Mathematics and Its Applications, Vol. 128 (Cambridge University Press, Cambridge, 2009). Crossref, Google Scholar
9. R. Belshoff and J. Chapman, Planar zero-divisor graphs, J. Algebra 316(1) (2007) 471–480. Crossref, Web of Science, Google Scholar
10. C. Berge, Graphs and Hypergraphs (Dunod, Paris, 1970). Google Scholar
11. C. Berge, Graphs and Hypergraphs (North-Holland Publishing Company, London, 2003). Google Scholar
12. G. Chartrand and L. Lesniak, Graphs and Digraphs (Wadsworth and Brooks/Cole, Monterey, 1986). Google Scholar
13. Ch. Eslahchi and A. M. Rahimi, The $k$ -zero divisor hypergraph of a commutative ring, Int. J. Math. Math. Sci. 2007 (2007) 50875. Crossref, Google Scholar
14. I. Kaplansky, Commutative Rings (University of Chicago, Chicago, 1974). Google Scholar
15. H. R. Maimani, C. Wickham and S. Yassemi, Rings whose total graphs have genus at most one, Rocky Mt. J. Math. 42 (2012) 1551–1560. Crossref, Web of Science, Google Scholar
16. B. Mohar and C. Thomassen, Graphs on Surfaces (The Johns Hopkins University Press, Baltimore, 1956). Google Scholar
17. A. R. Moniri Hamzekolaee and M. Norouzi, On characterization of finite modules by hypergraphs, An. St. Univ. Ovidius Constanta 30(1) (2022) 231–246. Google Scholar
18. A. R. Moniri Hamzekolaee and M. Norouzi, Applying a hypergraph to determine the structure of some finite modules, J. Appl. Math. Comput. 69 (2023) 675–687. Crossref, Google Scholar
19. K. Selvakumar and V. Ramanathan, Classification of non-local rings with genus one $3$ -zero-divisor hypergraphs, Commun. Algebra 45(1) (2017) 275–284. Crossref, Web of Science, Google Scholar
20. K. Selvakumar and V. Ramanathan, Classification of non-local rings with projective $3$ -zero-divisor hypergraph, Ric. Mat. 66 (2017) 457–468. Crossref, Web of Science, Google Scholar
21. N. O. Smith, Planar zero-divisor graphs, Int. J. Commut. Rings 2(4) (2003) 177–188. Google Scholar
22. T. Tamizh Chelvam, K. Selvakumar and V. Ramanathan, On the planarity of the $k$ -zero-divisor hypergraphs, AKCE Int. J. Graphs Comb. 12(2) (2015) 169–176. Crossref, Web of Science, Google Scholar
23. C. Thomassen, The graph genus problem is NP-complete, J. Algorithms 10 (1989) 568–576. Crossref, Web of Science, Google Scholar
24. T. R. S. Walsh, Hypermaps versus bipartite maps, J. Comb. Theory B 18 (1975) 155–163. Crossref, Web of Science, Google Scholar
25. H. J. Wang, Zero-divisor graphs of genus one, J. Algebra 304(2) (2006) 666–678. Crossref, Web of Science, Google Scholar
26. H. J. Wang and N. O. Smith, Commutative rings with toroidal zero-divisor graphs, Houston J. Math. 36(1) (2010) 1–31. Web of Science, Google Scholar
27. A. T. White, Graphs, Groups and Surfaces (North-Holland Publishing Company, Amsterdam, 1973). Google Scholar