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On the Total Graph of a Ring and Its Related Graphs: A Survey

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Commutative Algebra

Abstract

Let R be a (commutative) ring with nonzero identity and Z(R) be the set of all zero divisors of R. The total graph of R is the simple undirected graph T(Γ(R)) with vertices all elements of R, and two distinct vertices x and y are adjacent if and only if x + y ∈ Z(R). This type of graphs has been studied by many authors. In this paper, we state many of the main results on the total graph of a ring and its related graphs.

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References

  1. A. Abbasi, S. Habib, The total graph of a commutative ring with respect to proper ideals. J. Korean Math. Soc. 49, 85–98 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  2. S. Akbari, F. Heydari, The regular graph of a non-commutative ring. Bull. Austral. Math. Soc. 89, 132–140 (2013)

    Article  MathSciNet  Google Scholar 

  3. S. Akbari, M. Jamaali, S.A. Seyed Fakhari, The clique numbers of regular graphs of matrix algebras are finite. Linear Algebra Appl. 43, 1715–1718 (2009)

    Article  MathSciNet  Google Scholar 

  4. S. Akbari, D. Kiani, F. Mohammadi, S. Moradi, The total graph and regular graph of a commutative ring. J. Pure Appl. Algebra 213, 2224–2228 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. S. Akbari, M. Aryapoor, M. Jamaali, Chromatic number and clique number of subgraphs of regular graph of matrix algebras. Linear Algebra Appl. 436, 2419–2424 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  6. D.F. Anderson, A. Badawi, On the zero-divisor graph of a ring. Comm. Algebra 36, 3073–3092 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. D.F. Anderson, A. Badawi, The total graph of a commutative ring. J. Algebra 320, 2706–2719 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. D.F. Anderson, A. Badawi, The total graph of a commutative ring without the zero element. J. Algebra Appl. 11, (18 pages) (2012). doi:10.1142/S0219498812500740

    Google Scholar 

  9. D.F. Anderson, A. Badawi, The generalized total graph of a commutative ring. J. Algebra Appl. 12, (18 pages) (2013). doi:10.1142/S021949881250212X

    Google Scholar 

  10. D.F. Anderson, P.S. Livingston, The zero-divisor graph of a commutative ring. J. Algebra 217, 434–447 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  11. D.F. Anderson, S.B. Mulay, On the diameter and girth of a zero-divisor graph. J. Pure Appl. Algebra 210, 543–550 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. D.D. Anderson, M. Winders, Idealization of a module. J. Comm. Algebra, 1 3–56 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. D.F. Anderson, M. Axtell, J. Stickles, Zero-divisor graphs in commutative rings, in Commutative Algebra Noetherian and Non-Noetherian Perspectives, ed. by M. Fontana, S.E. Kabbaj, B. Olberding, I. Swanson (Springer, New York, 2010), pp. 23–45

    Google Scholar 

  14. D.F. Anderson, J. Fasteen, J.D. LaGrange, The subgroup graph of a group. Arab. J. Math. 1, 17–27 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  15. N. Ashra, H.R. Maimani, M.R. Pournaki, S. Yassemi, Unit graphs associated with rings. Comm. Algebra 38, 2851–2871 (2010)

    Article  MathSciNet  Google Scholar 

  16. S.E. Atani, S. Habibi, The total torsion element graph of a module over a commutative ring. An. Stiint. Univ.‘Ovidius Constanta Ser. Mat. 19, 23–34 (2011)

    Google Scholar 

  17. M. Axtel, J. Stickles: Zero-divisor graphs of idealizations. J. Pure Appl. Algebra 204, 23–43 (2006)

    Google Scholar 

  18. Z. Barati, K. Khashyarmanesh, F. Mohammadi, K. Nafar, On the associated graphs to a commutative ring. J. Algebra Appl. 11, (17 pages) (2012). doi:10.1142/S021949881105610

    Google Scholar 

  19. I. Beck, Coloring of commutative rings. J. Algebra 116, 208–226 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  20. B. Bollaboás, Graph Theory, An Introductory Course (Springer, New York, 1979)

    Google Scholar 

  21. T. Chelvam, T. Asir, Domination in total graph on \(\mathbb{Z}_{n}\). Discrete Math. Algorithms Appl. 3, 413–421 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  22. T. Chelvam, T. Asir, Domination in the total graph of a commutative ring. J. Combin. Math. Combin. Comput. (to appear)

    Google Scholar 

  23. T. Chelvam, T. Asir, Intersection graph of gamma sets in the total graph. Discuss. Math. Graph Theory 32, 339–354 (2012)

    Google Scholar 

  24. T. Chelvam, T. Asir, On the genus of the total graph of a commutative ring. Comm. Algebra 41, 142–153 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  25. T. Chelvam, T. Asir, On the total graph and its complement of a commutative ring. Comm. Algebra 41, 3820–3835 (2013). doi:10.1080/00927872.2012.678956

    Article  MATH  MathSciNet  Google Scholar 

  26. T. Chelvam, T. Asir, The intersection graph of gamma sets in the total graph I. J. Algebra Appl. 12, (18 pages), (2013). doi:10.1142/S0219498812501988

    Google Scholar 

  27. T. Chelvam, T. Asir, The intersection graph of gamma sets in the total graph II. J. Algebra Appl. 12, (14 pages), (2013). doi:10.1142/S021949881250199X

    Google Scholar 

  28. S. Endo, Note on p.p. rings. Nagoya Math. J. 17, 167–170 (1960)

    Google Scholar 

  29. J.A. Huckaba, Commutative Rings with Zero Divisors (Dekker. New York/Basel, 1988)

    Google Scholar 

  30. I. Kaplansky, Commutative Rings (University of Chicago Press, Chicago, 1974)

    MATH  Google Scholar 

  31. K. Khashyarmanesh, M.R. Khorsandi, A generalization of the unit and unitary Cayley graphs of a commutative ring. Acta Math. Hungar. 137, 242–253 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  32. H.R. Maimani, M.R. Pouranki, A. Tehranian, S. Yassemi, Graphs attached to rings revisited. Arab. J. Sci. Eng. 36, 997–1011 (2011)

    Article  MathSciNet  Google Scholar 

  33. H.R. Maimani, C. Wickham, S. Yassemi, Rings whose total graphs have genus at most one. Rocky Mountain J. Math. 42, 1551–1560 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  34. W.W. McGovern, Clean semiprime f-rings with bounded inversion. Comm. Algebra 31, 3295–3304 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  35. Z. Pucanović, Z. Petrović, On the radius and the relation between the total graph of a commutative ring and its extensions. Publ. Inst. Math. (Beograd)(N.S.) 89, 1–9 (2011)

    Google Scholar 

  36. P.K. Sharma, S.M. Bhatwadekar, A note on graphical representations of rings. J. Algebra 176, 124–127 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  37. M.H. Shekarriz, M.H. Shiradareh Haghighi, H. Sharif, On the total graph of a finite commutative ring. Comm. Algebra 40, 2798–2807 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, American University of Sharjah, 26666, Sharjah, United Arab Emirates

    Ayman Badawi

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  1. Ayman Badawi
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Correspondence to Ayman Badawi .

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Editors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi Roma Tre, Rome, Italy

    Marco Fontana

  2. Department of Mathematics, Graz University of Technology, Graz, Austria

    Sophie Frisch

  3. Department of Mathematics, University of Connecticut, Storrs, Connecticut, USA

    Sarah Glaz

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Badawi, A. (2014). On the Total Graph of a Ring and Its Related Graphs: A Survey. In: Fontana, M., Frisch, S., Glaz, S. (eds) Commutative Algebra. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0925-4_3

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