Journal of the
Korean Mathematical Society JKMS

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 $\mathbb{AG}(R)$.

Keywords: annihilating-ideal graph, graph blow-up, strong Boolean graph, complemented graph, pre-atomic graph, clique number

MSC numbers: 05C25, 05C75, 13A15