Abstract
Using the new extension of the zero-divisor graph \(\widetilde{\Gamma }(R)\), we give a new approach to the diameter of \(\Gamma (R)\) and \(\Gamma (R[X])\), other than that given by Lucas in his paper on the diameter of a zero divisor graph. We give a complete characterization for the possible diameters 1, 2 or 3 of \(\Gamma (R)\) and \(\Gamma (R[x])\).
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References
Anderson, D.D.: commutative rings. In: Multiplicative Ideal Theory in Commutative Algebra: A Tribute to the Work of Robert Gilmer. Springer, pp. 1–20 (2006)
Anderson, D.F., Badawi, A.: The total graph of a commutative ring. J. Algebra 320(7), 2706–2719 (2008)
Anderson, D.F., Livingston, P.S.: The zero-divisor graph of a commutative ring. J. Algebra 217, 434–447 (1999)
Atiyah, M.F., Macdonald, I.G.: Introduction to Commutative Algebra. Addison-Wesley Publishing Company (1969)
Beck, I.: Coloring of commutative rings. J. Algebra 116, 208–226 (1988)
Cherrabi, A., Essannouni, H., Jabbouri, E., Ouadfel, A.: On a new extension of the zero-divisor graph. Algebra Colloq. 27(3), 469–476 (2020)
Cherrabi, A., Essannouni, H., Jabbouri, E., Ouadfel, A.: On a new extension of the zero-divisor graph (II). Beitr. Zur Algebra Geom. 62(4), 945–953 (2021)
Diestel, R.: Graph Theory. Springer, New York (1997)
Huckaba, J.: Commutative Rings with Zero-divisors. Dekker, New York (1988)
Kaplansky, I.: Commutative Rings. University of Chicago (1974)
Lucas, T.G.: The diameter of zero-divisor graph. J. Algebra 301, 174–193 (2006)
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Cherrabi, A., Essannouni, H., Jabbouri, E. et al. A new approach to the diameter of zero-divisor graph. Beitr Algebra Geom 64, 545–554 (2023). https://doi.org/10.1007/s13366-022-00647-x
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DOI: https://doi.org/10.1007/s13366-022-00647-x