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RISS 인기검색어
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- 저자 : Houyi Yu (검색결과 2 건)
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On rings whose annihilating-ideal graphs are blow-ups of a class of Boolean graphs
Jin Guo,Tongsuo Wu,Houyi Yu 대한수학회 2017 대한수학회지 Vol.54 No.3
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 $2^X\setminus \{X,\,\eset\}$, with $M$ adjacent to $N$ if $M\cap N=\eset.$ In this paper, we characterize the annihilating-ideal graphs $\mathbb{AG}(R) $ that are blow-ups of strong Boolean graphs, complemented graphs and pre-atomic graphs respectively. In particular, for a commutative ring $R$ such that $\mathbb{AG}(R)$ has a maximum clique $S$ with $3\le |V(S)| \leq \infty$, 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 $\mathbb{AG}(R)$.
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ON RINGS WHOSE ANNIHILATING-IDEAL GRAPHS ARE BLOW-UPS OF A CLASS OF BOOLEAN GRAPHS
Guo, Jin,Wu, Tongsuo,Yu, Houyi Korean Mathematical Society 2017 대한수학회지 Vol.54 No.3
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 $\mathbb{AG}(R)$.
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