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A Characterization of Bipartite Zero-divisor Graphs

  • Nader Jafari Rad (a1) and Sayyed Heidar Jafari (a1)
Abstract

In this paper we obtain a characterization for all bipartite zero-divisor graphs of commutative rings R with 1 such that R is finite or |Nil(R)| ≠ 2.

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References
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[1] Akbari, S. and Mohammadian, A., Zero-divisor graphs of non-commutative rings. J. Algebra 296 (2006), 462479. http://dx.doi.org/10.1016/j.jalgebra.2005.07.007
[2] Akbari, S., Maimani, H. R., and S. Yassemi, When a zero-divisor graph is planar or a complete r-partitegraph. J. Algebra 270 (2003), 169180. http://dx.doi.org/10.1016/S0021-8693(03)00370-3
[3] Anderson, D. F., Frazier, A., Lauve, A., and Livingston, P. S., The zero-divisor graph of a commutativering, II. In: Ideal Theoretic Methods in Commutative Algebra (Columbia, MO, 1999), Dekker, New York, 2001, 6172.
[4] Anderson, D. F. and Livingston, P. S., The zero-divisor graph of a commutative ring. J. Algebra 217 (1999), 434447. http://dx.doi.org/10.1006/jabr.1998.7840
[5] Atiyah, M. F. and Macdonald, Ian G., Introduction to Commutative Algebra. Addison-Wesley Publishing Co, Reading, Mass.–London–Don Mills, Ont., 1969.
[6] Dancheng, L. andTongsuo, W., On bipartite zero-divisor graphs. Discrete Math. 309 (2009), 755762. http://dx.doi.org/10.1016/j.disc.2008.01.044
[7] DeMeyer, F. and Schneider, K., Automorphisms and zero divisor graphs of commutative rings. In: Commutative rings, Nova Sci. Publ., Hauppauge, NY, 2002, pp. 2537.
[8] Redmond, S. P., An ideal-based zero-divisor graph of a commutative ring. Comm. Algebra 31 (2003), 44254443. http://dx.doi.org/10.1081/AGB-120022801
[9] Singh, S. and Zameeruddin, Q., Modern Algebra. Third reprint, Vikas Publishing House Pvt. Ltd., Dehli, 1995.
[10] West, D. B., Introduction To Graph Theory. Prentice-Hall of India Pvt. Ltd, 2003.
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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