On annihilator graph of a finite commutative ring

Document Type: Research Paper

Authors

1 North eastern Hill University

2 North Eastern Hill University

Abstract

‎The annihilator graph $AG(R)$ of a commutative ring $R$ is a simple undirected graph with the vertex set $Z(R)^*$ and two distinct vertices are adjacent if and only if $ann(x) \cup ann(y)$ $ \neq $ $ann(xy)$‎. ‎In this paper we give the sufficient condition for a graph $AG(R)$ to be complete‎. ‎We characterize rings for which $AG(R)$ is a regular graph‎, ‎we show that $\gamma (AG(R))\in \{1,2\}$ and we also characterize the rings for which $AG(R)$ has a cut vertex‎. ‎Finally we find the clique number of a finite reduced ring and characterize the rings for which $AG(R)$ is a planar graph‎.

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[1] D. F. Anderson, A. Frazier, A. Lauve and P. S. Livingston, The zero-divisor graph of a commutative ring. II, Ideal theoretic metho ds in commutative algebra (Columbia, MO,1999), Lecture Notes in Pure and Appl. Math., 220, Dekker, New York, 2001 61-72.

[2] D. F. Anderson and P. S. Livingston, The Zero-divisor graph of a commutative ring, J. Algebra, 217 no. 2 (1999) 434-447.

[3] S. Akbari, H. R. Maimani and S. Yassemi, When a zero-divisor graph is planar or a complete r -partite graph, J. Algebra, 270 no. 1 (2003) 169-180.

[4] M. F. Aitiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont, 1969.

[5] S. Akbari and A. Mohammadian, On the zero-divisor graph of a commutative ring, J.Algebra, 274 no. 2 (2004) 847-855.

[6] M. Axtell, J. Stickles and W. Trampbachls, Zero-divisor ideals and realizable zero-divisor graphs, Involve, 2 no. 1 (2009) 17-27.

[7] A. Badawi, On the annihilator graph of a commutative ring, Comm. Algebra, 42 no. 1 (2014) 108-121.

[8] I. Beck, Coloring of commutative rings, J. Algebra, 116 no. 1 (1998) 208-226.

[9] T. T. Chelvam and T. Asir, Domination in the Total Graph on Zn , Discrete Math. Algorithms Appl., 3 no. 4 (2011) 413-421.

[10] D. A. Mo jdeh and A. M. Rahimi, Dominating Sets of Some Graphs Asso ciated to Commutative Rings, Comm. Algebra, 40 no. 9 (2012) 3389-3396.