Revista de la
Unión Matemática Argentina
On the planarity, genus, and crosscap of the weakly zero-divisor graph of commutative rings
Nadeem ur Rehman, Mohd Nazim, and Shabir Ahmad Mir

Volume 67, no. 1 (2024), pp. 213–227    

Published online: April 30, 2024

https://doi.org/10.33044/revuma.2837

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Abstract

Let $R$ be a commutative ring and $Z(R)$ its zero-divisors set. The weakly zero-divisor graph of $R$, denoted by $W\Gamma(R)$, is an undirected graph with the nonzero zero-divisors $Z(R)^*$ as vertex set and two distinct vertices $x$ and $y$ are adjacent if and only if there exist $a \in \mathrm{Ann}(x)$ and $b \in \mathrm{Ann}(y)$ such that $ab = 0$. In this paper, we characterize finite rings $R$ for which the weakly zero-divisor graph $W\Gamma(R)$ belongs to some well-known families of graphs. Further, we classify the finite rings $R$ for which $W\Gamma(R)$ is planar, toroidal or double toroidal. Finally, we classify the finite rings $R$ for which the graph $W\Gamma(R)$ has crosscap at most two.

References

  1. S. Akbari and A. Mohammadian, On the zero-divisor graph of a commutative ring, J. Algebra 274 no. 2 (2004), 847–855.  DOI  MR  Zbl
  2. D. D. Anderson and M. Naseer, Beck's coloring of a commutative ring, J. Algebra 159 no. 2 (1993), 500–514.  DOI  MR  Zbl
  3. D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 no. 2 (1999), 434–447.  DOI  MR  Zbl
  4. T. Asir and K. Mano, Classification of rings with crosscap two class of graphs, Discrete Appl. Math. 265 (2019), 13–21.  DOI  MR  Zbl
  5. T. Asir and K. Mano, Classification of non-local rings with genus two zero-divisor graphs, Soft Comput. 24 no. 1 (2020), 237–245, correction ibid. 25, no. 4, 3355–3356 (2021).  DOI  Zbl
  6. M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley, Reading, Mass.-London-Don Mills, Ont., 1969.  MR  Zbl
  7. I. Beck, Coloring of commutative rings, J. Algebra 116 no. 1 (1988), 208–226.  DOI  MR  Zbl
  8. N. Bloomfield and C. Wickham, Local rings with genus two zero divisor graph, Comm. Algebra 38 no. 8 (2010), 2965–2980.  DOI  MR  Zbl
  9. H.-J. Chiang-Hsieh, Classification of rings with projective zero-divisor graphs, J. Algebra 319 no. 7 (2008), 2789–2802.  DOI  MR  Zbl
  10. F. R. DeMeyer, T. McKenzie, and K. Schneider, The zero-divisor graph of a commutative semigroup, Semigroup Forum 65 no. 2 (2002), 206–214.  DOI  MR  Zbl
  11. I. Gitler, E. Reyes, and R. H. Villarreal, Ring graphs and complete intersection toric ideals, Discrete Math. 310 no. 3 (2010), 430–441.  DOI  MR  Zbl
  12. B. R. McDonald, Finite rings with identity, Pure and Applied Mathematics, Vol. 28, Marcel Dekker, New York, 1974.  MR  Zbl
  13. B. Mohar and C. Thomassen, Graphs on surfaces, Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 2001.  MR  Zbl
  14. M. J. Nikmehr, A. Azadi, and R. Nikandish, The weakly zero-divisor graph of a commutative ring, Rev. Un. Mat. Argentina 62 no. 1 (2021), 105–116.  DOI  MR  Zbl
  15. S. P. Redmond, The zero-divisor graph of a non-commutative ring, Int. J. Commut. Rings 1 no. 4 (2002), 203–211.  Zbl
  16. H.-J. Wang, Zero-divisor graphs of genus one, J. Algebra 304 no. 2 (2006), 666–678.  DOI  MR  Zbl
  17. D. B. West, Introduction to graph theory, 2nd ed., New Delhi: Prentice-Hall of India, 2005.  Zbl
  18. A. T. White, Graphs, groups and surfaces, North-Holland Mathematics Studies, No. 8, North-Holland, Amsterdam-London; American Elsevier, New York, 1973.  MR  Zbl
  19. C. Wickham, Classification of rings with genus one zero-divisor graphs, Comm. Algebra 36 no. 2 (2008), 325–345.  DOI  MR  Zbl