Geodesic
A variation of zero-divisor graphs
Discussiones Mathematicae. General Algebra and Applications, Tome 35 (2015) no. 2, pp. 159-176.

Voir la notice de l'article provenant de la source Library of Science

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.
Keywords: rings, zero-divisor graphs, finite fields 
@article{DMGAA_2015_35_2_a3,
     author = {Gupta, Raibatak and Sen, M. and Ghosh, Shamik},
     title = {A variation of zero-divisor graphs},
     journal = {Discussiones Mathematicae. General Algebra and Applications},
     pages = {159--176},
     publisher = {mathdoc},
     volume = {35},
     number = {2},
     year = {2015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGAA_2015_35_2_a3/}
}
TY  - JOUR
AU  - Gupta, Raibatak
AU  - Sen, M.
AU  - Ghosh, Shamik
TI  - A variation of zero-divisor graphs
JO  - Discussiones Mathematicae. General Algebra and Applications
PY  - 2015
SP  - 159
EP  - 176
VL  - 35
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGAA_2015_35_2_a3/
LA  - en
ID  - DMGAA_2015_35_2_a3
ER  - 
%0 Journal Article
%A Gupta, Raibatak
%A Sen, M.
%A Ghosh, Shamik
%T A variation of zero-divisor graphs
%J Discussiones Mathematicae. General Algebra and Applications
%D 2015
%P 159-176
%V 35
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGAA_2015_35_2_a3/
%G en
%F DMGAA_2015_35_2_a3
Gupta, Raibatak; Sen, M.; Ghosh, Shamik. A variation of zero-divisor graphs. Discussiones Mathematicae. General Algebra and Applications, Tome 35 (2015) no. 2, pp. 159-176. http://geodesic.mathdoc.fr/item/DMGAA_2015_35_2_a3/

[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).