Document Type: Research Paper


1 Yazd University

2 Payame Noor University


Let $G=(V,E)$ be a simple graph. A set $S\subseteq V$ is
independent set of $G$,  if no two vertices of $S$ are adjacent.
The  independence number $\alpha(G)$ is the size of a maximum
independent set in the graph.
In this paper we study and characterize the independent sets of
the zero-divisor graph $\Gamma(R)$ and ideal-based zero-divisor graph
of a commutative ring $R$.


[1] S. Akbari, H.R. Maimani, S. Yassemi, When a zero-divisor graph is planar or a complete r-partite graph, J. Algebra 270 (2003) 169-180.
[2] S. Akbari, A. Mohammadian, On the zero-divisor graph of a commutative ring, J. Algebra 274 (2004) 847-855.
[3] S. Alikhani and Y.H. Peng, Independence roots and independence fractals of certain graphs, J. Appl. Math. Com- puting, vol. 36, no. 1-2, (2011) 89-100.
[4] D.F. Anderson, A. Frazier, A. Lauve, P.S. Livingston, The zero-divisor graph of a commutative ring, II, in: Lecture Notes in Pure and Appl. Math., vol. 220, Dekker, New York, (2001) 61-72.
[5] D.F. Anderson, P.S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999) 434 447.
[6] M.F. Atiyah and I.G. MacDonald, Introduction to Commutative Algebra. Reading, MA: Addison-Wesley, 1969.
[7] M. Axtell, J. Coykendall, J. Stickles, Zero-divisor graphs of polynomials and power series over commutative rings, Comm. Algebra. 6 (2005) 2043-2050.
[8] I. Beck, Coloring of commutative rings, J. Algebra 116 (1988) 208-226.
[9] I. Gutman, F. Harary, Generalization of the matching polynomial, Utilitas Mathematica 24 (1983) 97-106.
[10] C. Hoede, and X. Li, Clique polynomials and independent set polynomials of graphs, Discrete Math. 25 (1994) 219-228.
[11] N. Jafari Rad, S. H. Jafari, D.A. Mojdeh, On domination in zero-divisor graphs, Canad. Math. Bull.
DOI:10.4153/CMB-2011-156-1 (2012).
[12] I. Kaplansky, Commutative Rings, Chicago-London: The University of Chicago Press, 1974.
[13] D. A. Mojdeh and A. M. Rahimi, Dominating Sets of Some Graphs Associated to Commutative Rings, Comm. Alg., 40:9 (2012) 3389-3396.
[14] S.P. Redmond, The zero-divisor graph of a non-commutative ring, Internat. J. Commutative Rings 1 (4) (2002) 203-211.
[15] S.P. Redmond, On zero-divisor graphs of small finite commutative rings, Discrete Math. 307 (2007) 1155–1166.
[16] S.P. Redmond, An ideal-based zero-divisor graph of a commutative ring, Comm. Alg. 31, (2003) 4425-4443.
[17] D.B. West, Introduction to Graph Theory, 2nd ed. USA: Prentice Hal, (2001).