Download PDF - Some properties of the zero divisor graph of a commutative ringAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Some properties of the zero divisor graph of a commutative ring

Authors 1, 2

Affiliations

  1. Department of Mathematics, Palestine Technical University-Kadoorie, Tulkarm, West Bank, Palestine
  2. Department of Mathematics, Jordan University, Amman 11942 Jordan

Abstract

Let Γ(R) be the zero divisor graph for a commutative ring with identity. The k-domination number and the 2-packing number of Γ(R), where R is an Artinian ring, are computed. k-dominating sets and 2-packing sets for the zero divisor graph of the ring of Gaussian integers modulo n, Γ(ℤₙ[i]), are constructed. The center, the median, the core, as well as the automorphism group of Γ(ℤₙ[i]) are determined. Perfect zero divisor graphs Γ(R) are investigated.

Keywords

ENG
automorphism group of a graph, center of a graph, core of a graph, k-domination number, Gaussian integers modulo n, median of a graph, 2-packing, perfect graph, and zero divisor graph

Bibliography

  1. E. Abu Osba, The Complement graph for Gaussian integers modulo n, Commun. Algebra 40 (5), (2012) 1886-1892. doi: 10.1080/00927872.2011.560588
  2. E. Abu Osba, S. Al-Addasi and N. Abu Jaradeh, Zero divisor graph for the ring of Gaussian integers modulo n, Commun. Algebra 36 (10) (2008) 3865-3877. doi: 10.1080/00927870802160859
  3. E. Abu Osba, S. Al-Addasi and B. Al-Khamaiseh, Some properties of the zero divisor graph for the ring of Gaussian integers modulo n, Glasgow J. Math. 53 (1) (2011) 391-399. doi: 10.1017/S0017089511000024
  4. S. Akbari and A. Mohamamadaian, On the zero divisor graph of a commutative ring, J. Algebra 274 (2004) 847-855. doi: 10.1016/S0021-8693(03)00435-6
  5. D.F. Anderson, M.C. Axtell and J.A. Stickles, Zero-divisor graphs in commutative rings, Commutative Algebra in Noetherian and Non-Noetherian Perspectives (M. Fontana, S.-E. Kabbaj, B. Olberding, I. Swanson, Eds.), Springer-Verlag, New York (2011), 2345.
  6. D.F. Anderson and A.D. Badawi, On the zero-divisor graph of a ring, Commun. Algebra 36 (2008) 3073-3092. doi: 10.1080/0092787080211088
  7. D.F. Anderson, A. Frazier, A. Lauve and P.S. Livingston, The zero divisor graph of a commutative ring II, Lecture notes in Pure and Appl. Math., New Yourk, Marcel Dekker 220 (2001) 61-72.
  8. D.F. Anderson and P.S. Livingston, The zero divisor graph of a commutative ring, J. Algebra 217 (1999) 434-447. doi: 10.1006/jabr.1998.7840
  9. S. Arumugam and S. Velammal, Edge domination in graphs, Taiwanese J. Math. 2 (2) (1998) 173-179.
  10. V.K. Bahat and R. Raina, A note on zero divisor graph over rings, Int. J. Contemp. Math. Sci. 14 (2) (2007) 667-671.
  11. J. Beck, Coloring of Commutative rings, J. Algebra 116 (1988) 208-226. doi: 10.1016/0021-8693(88)90202-5
  12. C. Berge, Fäbung von Graphen, deren s Fäamtliche bzw. Deren ungerade Kreise starr sind, Wiss, Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe (1961) 114-115.
  13. G. Chartrand and L. Leśniak, Graphs and Digraphs, 2 ed., Wadsworth and Brooks (Monterey, California, 1986).
  14. H. Chiang-Hsieh, H. Wang and N. Smith, Commutative rings with toroidal zero-divisor graphs, Houston J. Math. 36 (1) (2010) 1-31.
  15. M. Chudnovsky, N. Robertson, P. Seymour and R. Thomas, The strong perfect graph theorem, Ann. Math. 164 (1) (2006) 51-229. doi: 10.4007/annals.2006.164.51
  16. N. Cordova, C. Gholston and H. Hauser, The Structure of Zero-divisor Graphs (Sumsri, Miami University, 2005).
  17. M. El-Zahar and C. Pareek, Domination number of products of graphs, Ars Combin. 31 (1991) 223-227.
  18. M. Ghanem and K. Nazzal, On the line graph of the complement graph for the ring of Gaussian integers modulo n, Open J. Disc. Math. 2 (1) (2012) 24-34.
  19. T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamental of Domination in Graphs (Marcel Dekker, New York, 1998).
  20. T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in Graphs, Advanced Topics, Marcel Dekker, Inc. (New York, 1998).
  21. R.B. Hayward, Murky graphs, J. Combin. Theory (B) 49 (1990) 200-235.
  22. Ph.S. Livingston, Structure in Zero-Divisor Graphs of Commutative Rings, Masters Theses, University of Tennessee (Knoxville, 1997).
  23. L. Lovász, Normal hypergraphs and the perfect graph conjecture, Disc. Math. 2 (1972) 253-267.
  24. T.F. Maire, Slightly triangulated graphs are perfect, Graphs Combin. 10 (1994) 263-268.
  25. K. Nazzal and M. Ghanem, On the line graph of the zero divisor graph for the ring of Gaussian integers modulo n, Int. J. Comb. (2012) 13 pages.
  26. S.P. Redmond, Central sets and Radii of the zero divisor graphs of commutative ring, Commun. Algebra 34 (2006) 2389-2401.
Discussiones Mathematicae - General Algebra and Applications
2014/34/2
Export to PBN, Opens in new tab
Pages:
167-181
Main language of publication
English
Published
2014

DOI: 10.7151/dmgaa.1222

Exact and natural sciences