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

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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.

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  1. 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

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  3. 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 

  4. 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

    Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  6. 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 

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

    MathSciNet  Article  Google Scholar 

  8. 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

    MathSciNet  Article  Google Scholar 

  9. 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 

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

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

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

    Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  15. 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 

  16. 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

    MathSciNet  Article  Google Scholar 

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

    MATH  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

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

    Google Scholar 

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

    MathSciNet  Article  Google Scholar 

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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).

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Correspondence to T. Asir.

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Asir, T., Mano, K. Classification of non-local rings with genus two zero-divisor graphs. Soft Comput 24, 237–245 (2020).

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  • Non-local rings
  • Zero-divisor graphs
  • Genus of a graph
  • Double torus