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ON RINGS WHOSE ANNIHILATING-IDEAL GRAPHS ARE BLOW-UPS OF A CLASS OF BOOLEAN GRAPHS

  • Guo, Jin (College of Information Science and Technology Hainan University) ;
  • Wu, Tongsuo (Department of Mathematics Shanghai Jiaotong University) ;
  • Yu, Houyi (School of Mathematics and Statistics Southwest University)
  • Received : 2016.04.23
  • Published : 2017.05.01

Abstract

For a finite or an infinite set X, let $2^X$ be the power set of X. A class of simple graph, called strong Boolean graph, is defined on the vertex set $2^X{\setminus}\{X,{\emptyset}\}$, with M adjacent to N if $M{\cap}N={\emptyset}$. In this paper, we characterize the annihilating-ideal graphs $\mathbb{AG}(R)$ that are blow-ups of strong Boolean graphs, complemented graphs and preatomic graphs respectively. In particular, for a commutative ring R such that AG(R) has a maximum clique S with $3{\leq}{\mid}V(S){\mid}{\leq}{\infty}$, we prove that $\mathbb{AG}(R)$ is a blow-up of a strong Boolean graph if and only if it is a complemented graph, if and only if R is a reduced ring. If assume further that R is decomposable, then we prove that $\mathbb{AG}(R)$ is a blow-up of a strong Boolean graph if and only if it is a blow-up of a pre-atomic graph. We also study the clique number and chromatic number of the graph $\mathbb{AG}(R)$.

Keywords

annihilating-ideal graph;graph blow-up;strong Boolean graph;complemented graph;pre-atomic graph;clique number

Acknowledgement

Supported by : National Natural Science Foundation of China, Hainan Natural Science Foundation

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