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2015 | 35 | 2 | 159-176
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A variation of zero-divisor graphs

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Warianty tytułu
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EN
Abstrakty
EN
In this paper, we define a new graph for a ring with unity by extending the definition of the usual 'zero-divisor graph'. For a ring R with unity, Γ₁(R) is defined to be the simple undirected graph having all non-zero elements of R as its vertices and two distinct vertices x,y are adjacent if and only if either xy=0 or yx=0 or x+y is a unit. We consider the conditions of connectedness and show that for a finite commutative ring R with unity, Γ₁(R) is connected if and only if R is not isomorphic to ℤ₃ or $ℤ₂^k$ (for any k ∈ ℕ-{1}\). Then we characterize the rings R for which Γ₁(R) realizes some well-known classes of graphs, viz., complete graphs, star graphs, paths (i.e., $P_n$), or cycles (i.e., $C_n$). We then look at different graph-theoretical properties of the graph Γ₁(F), where F is a finite field. We also find all possible Γ₁(R) graphs with at most 6 vertices.
Słowa kluczowe
Twórcy
  • Department of Mathematics, Jadavpur University, Kolkata - 700032, India
autor
  • Department of Pure Mathematics, University of Calcutta, 35, Ballygunge Circular Road, Kolkata - 700019, India
autor
  • Department of Mathematics, Jadavpur University, Kolkata - 700032, India
Bibliografia
  • [1] D.D. Anderson and M. Naseer, Beck's coloring of a commutative ring, J. Algebra 159 (1993), 500-514. doi: 10.1006/jabr.1993.1171
  • [2] D.F. Anderson, M.C. Axtell and J.A. Stickles Jr., Zero-divisor graphs in commutative rings, Commutative Algebra: Noetherian and Non-Noetherian Perspectives (2011), 23-45. doi: 10.1007/978-1-4419-6990-3_2
  • [3] D.F. Anderson and P. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999), 434-447. doi: 10.1006/jabr.1998.7840
  • [4] 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., Marcel Dekker, New York 220 (2001), 61-72.
  • [5] N. Ashrafi, H.R. Maimani, M.R. Pournaki and S. Yassemi, Unit graphs associated with rings, Comm. Algebra 38 (2010), 2851-2871. doi: 10.1080/00927870903095574
  • [6] S.E. Atani, M.S. Kohan and Z.E. Sarvandi, An ideal-based zero-divisor graph of direct products of commutative rings, Discuss. Math. Gen. Algebra Appl. 34 (2014), 45-53. doi: 10.7151/dmgaa.1211
  • [7] M. Axtell, J. Stickles and W. Trampbachls, Zero-divisor ideals and realizable zero-divisor graphs, Involve 2 (2009), 17-27. doi: 10.2140/involve.2009.2.17
  • [8] I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), 208-226. doi: 10.1016/0021-8693(88)90202-5
  • [9] I. Bozic and Z. Petrovic, Zero-divisor graphs of matrices over commutative rings, Comm. Algebra 37 (2009), 1186-1192. doi: 10.1080/00927870802465951
  • [10] N. Ganesan, Properties of rings with a finite number of zero divisors, Math. Annalen 157 (3) (1964), 215-218. doi: 10.1007/BF01362435
  • [11] S. Redmond, The zero-divisor graph of a non-commutative ring, International Journal of Commutative Rings 1 (4) (2002), 203-211.
  • [12] D.B. West, Introduction to Graph Theory (Prentice Hall of India New Delhi, 2003).
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1238
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