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Perfect unit graphs of commutative Artinian rings

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Abstract

Let R be a commutative ring with identity. The unit graph of R, denoted by G(R), has its set of vertices equal to the set of all elements of R and two distinct vertices x and y are adjacent if and only if \(x+y\) is a unit of R. In this paper, perfect unit graphs of Artinian rings are investigated.

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References

  1. Akbari, S., Estaji, E., Khorsandi, M.R.: On the unit graph of a non-commutative ring. Algebra Colloq. 817, 817–822 (2015)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Atiyah, M.F., Macdonald, I.G.: Introduction to commutative algebra. Addison-Wesley Publishing Company, Boston (1969)

    MATH  Google Scholar 

  4. Bruns, W., Herzog, J.: Cohen-macaulay rings. Cambridge University Press, Cambridge (1997)

    MATH  Google Scholar 

  5. Das, A.K., Maimani, H.R., Pournaki, M.R., Yassemi, S.: Nonplanarity of unit graphs and classiffication of the toroidal ones. Pacific J. Math. 268, 371–387 (2014)

    Article  MathSciNet  Google Scholar 

  6. Diestel, R.: Graph Theory. Springer-Verlag, New York (2000)

    MATH  Google Scholar 

  7. Grimaldi, R.P.: Graphs from rings, in proceedings of the twentieth southeastern conference on combinatorics, graph theory, and computing (Boca Raton, FL, 1989). Congr. Numer. 71, 95–103 (1990)

    MathSciNet  Google Scholar 

  8. Heydari, F., Nikmehr, M.J.: The unit graph of a left Artinian ring. Acta Math. Hungarica 139, 134–146 (2013)

    Article  MathSciNet  Google Scholar 

  9. Maimani, H.R., Pournaki, M.R., Yassemi, S.: A class of weakly perfect graphs. Czech. Math. J. 60, 1037–1041 (2010)

    Article  MathSciNet  Google Scholar 

  10. Maimani, H.R., Pournaki, M.R., Yassemi, S.: Weakly perfect graphs arising from rings. Glasgow Math. J. 52, 417–425 (2010)

    Article  MathSciNet  Google Scholar 

  11. Goldsmith, B., Pabst, S., Scott, A.: Unit sum numbers of rings and modules. Quart. J. Math. Oxford Ser. 49, 331–344 (1998)

    Article  MathSciNet  Google Scholar 

  12. Su, H., Noguchi, K., Zhou, Y.: Finite commutative rings with higher genus unit graphs. J. Algebra Appl. 14, 1550002 (2015)

    Article  MathSciNet  Google Scholar 

  13. Su, H., Zhou, Y.: On the girth of the unit graph of a ring. J. Algebra Appl. 13, 1350082 (2014)

    Article  MathSciNet  Google Scholar 

  14. West, D.B.: Introduction to graph theory, 2nd edn. Prentice Hall, Upper Saddle River (2001)

    Google Scholar 

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Acknowledgements

The authors express their deep gratitude to the referees for their valuable suggestions which have definitely improved the paper.

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Authors and Affiliations

  1. Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran

    S. M. Saadat Mirghadim

  2. Department of Mathematics, Jundi-Shapur University of Technology, Dezful, Iran

    R. Nikandish

  3. Faculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran

    M. J. Nikmehr

Authors
  1. S. M. Saadat Mirghadim
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  2. R. Nikandish
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  3. M. J. Nikmehr
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Correspondence to R. Nikandish.

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Mirghadim, S.M.S., Nikandish, R. & Nikmehr, M.J. Perfect unit graphs of commutative Artinian rings. Afr. Mat. 32, 891–896 (2021). https://doi.org/10.1007/s13370-020-00868-0

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  • DOI: https://doi.org/10.1007/s13370-020-00868-0

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