On the Cozero-Divisor Graphs of Commutative Rings
ABSTRACT
Let R be a commutative ring with non-zero identity. The cozero-divisor graph of R, denoted by
, is a graph with vertices in 
, which is the set of all non-zero and non-unit elements of R, and two distinct vertices a and b in 
are adjacent if and only if 
and 
. In this paper, we investigate some combinatorial properties of the cozero-divisor graphs 
and 
such as connectivity, diameter, girth, clique numbers and planarity. We also study the cozero-divisor graphs of the direct products of two arbitrary commutative rings.
Let R be a commutative ring with non-zero identity. The cozero-divisor graph of R, denoted by
, is a graph with vertices in , which is the set of all non-zero and non-unit elements of R, and two distinct vertices a and b in are adjacent if and only if and . In this paper, we investigate some combinatorial properties of the cozero-divisor graphs and such as connectivity, diameter, girth, clique numbers and planarity. We also study the cozero-divisor graphs of the direct products of two arbitrary commutative rings.
Cite this paper
M. Afkham and K. Khashyarmanesh, "On the Cozero-Divisor Graphs of Commutative Rings," Applied Mathematics, Vol. 4 No. 7, 2013, pp. 979-985. doi: 10.4236/am.2013.47135.
M. Afkham and K. Khashyarmanesh, "On the Cozero-Divisor Graphs of Commutative Rings," Applied Mathematics, Vol. 4 No. 7, 2013, pp. 979-985. doi: 10.4236/am.2013.47135.
References
[1] I. Beck, “Coloring of Commutative Rings,” Journal of Algebra, Vol. 116, No. 1, 1988, pp. 208-226. doi:10.1016/0021-8693(88)90202-5 [2] D. F. Anderson and P. S. Livingston, “The Zero-Divisor Graph of a Commutative Ring,” Journal of Algebra, Vol. 217, No. 2, 1999, pp. 434-447. doi:10.1006/jabr.1998.7840 [3] M. Axtell, J. Coykendall and J. Stickles, “Zero-Divisor Graphs of Polynomials and Power Series over Commutative Rings,” Communications in Algebra, Vol. 33, No. 6, 2005, pp. 2043-2050. doi:10.1081/AGB-200063357 [4] T. G. Lucas, “The Diameter of a Zero Divisor Graph,” Journal of Algebra, Vol. 301, No. 1, 2006, pp. 174-193. doi:10.1016/j.jalgebra.2006.01.019 [5] M. Axtell, J. Stickles and J. Warfel, “Zero-Divisor Graphs of Direct Products of Commutative Rings,” Houston Journal of Mathematics, Vol. 32, No. 4, 2006, pp. 985-994. [6] M. Afkhami and K. Khashyarmanesh, “The Cozero-Divisor Graph of a Commutative Ring,” Southeast Asian Bulletin of Mathematics, Vol. 35, No. 5, 2011, pp. 753762. [7] G. Chartrand and O. R. Oellermann, “Applied and Algorithmic Graph Theory,” McGraw-Hill, Inc., New York, 1993. [8] J. A. Bondy and U. S. R. Murty, “Graph Theory with Applications,” American Elsevier, New York, 1976.
[1] I. Beck, “Coloring of Commutative Rings,” Journal of Algebra, Vol. 116, No. 1, 1988, pp. 208-226. doi:10.1016/0021-8693(88)90202-5 [2] D. F. Anderson and P. S. Livingston, “The Zero-Divisor Graph of a Commutative Ring,” Journal of Algebra, Vol. 217, No. 2, 1999, pp. 434-447. doi:10.1006/jabr.1998.7840 [3] M. Axtell, J. Coykendall and J. Stickles, “Zero-Divisor Graphs of Polynomials and Power Series over Commutative Rings,” Communications in Algebra, Vol. 33, No. 6, 2005, pp. 2043-2050. doi:10.1081/AGB-200063357 [4] T. G. Lucas, “The Diameter of a Zero Divisor Graph,” Journal of Algebra, Vol. 301, No. 1, 2006, pp. 174-193. doi:10.1016/j.jalgebra.2006.01.019 [5] M. Axtell, J. Stickles and J. Warfel, “Zero-Divisor Graphs of Direct Products of Commutative Rings,” Houston Journal of Mathematics, Vol. 32, No. 4, 2006, pp. 985-994. [6] M. Afkhami and K. Khashyarmanesh, “The Cozero-Divisor Graph of a Commutative Ring,” Southeast Asian Bulletin of Mathematics, Vol. 35, No. 5, 2011, pp. 753762. [7] G. Chartrand and O. R. Oellermann, “Applied and Algorithmic Graph Theory,” McGraw-Hill, Inc., New York, 1993. [8] J. A. Bondy and U. S. R. Murty, “Graph Theory with Applications,” American Elsevier, New York, 1976.