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http://dx.doi.org/10.4134/JKMS.j160283

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)
Publication Information
Journal of the Korean Mathematical Society / v.54, no.3, 2017 , pp. 847-865 More about this Journal
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;
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