Skip to main content
SpringerLink
Log in

Classification of non-local rings with genus two zero-divisor graphs

  • Foundations
  • Published:
Soft Computing Aims and scope Submit manuscript

A Correction to this article was published on 30 January 2021

This article has been updated

Abstract

The zero-divisor graph of a commutative ring R is a simple graph whose vertices are the nonzero zero divisors of R and two distinct vertices are adjacent if their product is zero. In this article, we determine precisely all non-local commutative rings whose zero-divisor graphs have genus two.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Change history

References

  • Akbari S, Maimani HR, Yassemi S (2003) When a zero-divisor graph is planar or a complete $r$-partite graph. J Algebra 270(1):169–180

    Article  MathSciNet  Google Scholar 

  • Anderson DF, Livingston PS (1999) The zero-divisor graph of a commutative ring. J Algebra 217:434–447

    Article  MathSciNet  Google Scholar 

  • Anderson DF, Frazier A, Lauve A, Livingston PS (2001) The zero-divisor graph of a commutative ring, II. Lect Notes Pure Appl Math 220:61–72

    MathSciNet  MATH  Google Scholar 

  • Anderson DF, Axtell MC, Stickles JA (2011) Zero-divisor graphs in commutative rings. Commutative algebra: Noetherian and Non-Noetherian. Springer, New York, pp 23–46

    MATH  Google Scholar 

  • Asir T (2018) The genus two class of graphs arising from rings. J Algebra Appl 17(10):1850193

    Article  MathSciNet  Google Scholar 

  • Asir T, Mano K (2018) The classification of rings with its genus of class of graphs. Turk J Math 42(3):1424–1435

    MathSciNet  MATH  Google Scholar 

  • Asir T, Mano K (2019) Classification of rings with crosscap two class of graphs. Discrete Appl Math 256:13–21

    Article  MathSciNet  Google Scholar 

  • Asir T, Tamizh Chelvam T (2014) On the genus of generalized unit and unitary Cayley graphs of a commutative ring. Acta Math Hung 142(2):444–458

    Article  MathSciNet  Google Scholar 

  • Asir T, Tamizh Chelvam T (2018) On the genus two characterizations of unit, unitary Cayley and co-maximal graphs. Ars Comb 138:77–91

    MathSciNet  MATH  Google Scholar 

  • Beck I (1988) Coloring of commutative rings. J Algebra 116(1):208–226

    Article  MathSciNet  Google Scholar 

  • Belshoff R, Chapman J (2007) Planar zero-divisor graphs. J Algebra 316:471–480

    Article  MathSciNet  Google Scholar 

  • Bloomfield N, Wickham C (2010) Local rings with genus two zero-divisor graph. Commun Algebra 38:2965–2980

    Article  MathSciNet  Google Scholar 

  • Chen J (2004) Minimum and maximum imbeddings. In: Gross JL, Yellen J (eds) Handbook of graph theory. CRC Press, Boca Raton

    Google Scholar 

  • Chiang-Hsieh H-J (2008) Classification of rings with projective zero-divisor graphs. J Algebra 319(7):2789–2802

    Article  MathSciNet  Google Scholar 

  • Chiang-Hsieh H-J, Smith NO, Wang H-J (2010) Commutative rings with toroidal zero-divisor graphs. Houst J Math 36(1):1–31

    MathSciNet  MATH  Google Scholar 

  • Ringeisen RD (1972) Determining all compact orientable 2-manifolds upon which $K_{m, n}$ has 2-cell imbeddings. J Comb Theory B 12:101–104

    Article  MathSciNet  Google Scholar 

  • Smith NO (2003) Planar zero-divisor graphs. Int J Commut Rings 2(4):177–188

    MATH  Google Scholar 

  • Smith NO (2007) Infinite planar zero-divisor graphs. Commun Algebra 35(1):171–180

    Article  MathSciNet  Google Scholar 

  • Tamizh Chelvam T, Asir T (2013) On the genus of the total graph of a commutative ring. Commun Algebra 41:142–153

    Article  MathSciNet  Google Scholar 

  • Wang H-J (2006) Zero-divisor graphs of genus one. J Algebra 304:666–678

    Article  MathSciNet  Google Scholar 

  • White AT (1973) Graphs, groups and surfaces. Elsevier, Amsterdam

    MATH  Google Scholar 

  • Wickham C (2008) Classification of rings with genus one zero-divisor graphs. Commun Algebra 36(2):325–345

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The research of T. Asir was in part supported by a Grant from The Science and Engineering Research Board (SERB–MATRICS Project—Ref. MTR/2017/000830). The research of K. Mano was in part supported by a fellowship from The University Grants Commission (Rajiv Gandhi National Fellowship—F1-17.1/2014-15/RGNF-2014-15-SC-TAM-85000).

Author information

Authors and Affiliations

  1. Department of Mathematics-DDE, Madurai Kamaraj University, Madurai, Tamil Nadu, 625 021, India

    T. Asir & K. Mano

Authors
  1. T. Asir
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. Mano
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. Asir.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interests regarding the publishing of this paper.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Additional information

Communicated by A. Di Nola.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Asir, T., Mano, K. Classification of non-local rings with genus two zero-divisor graphs. Soft Comput 24, 237–245 (2020). https://doi.org/10.1007/s00500-019-04345-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-019-04345-0

Keywords

Search

Navigation