Annihilating-Ideal Graph of $C(X)$

Abstract

In this article we study the annihilating-ideal graph of the ring $C(X)$. We have tried to associate the graph properties of $\mathbb{AG}(X)$, the ring properties of $C(X)$ and the topological properties of $X$. We have shown that $X$ has an isolated point \ff $\mathbb{R}$ is a direct summand of $C(X)$ if and only if $\mathbb{AG}(X)$ is not triangulated. Radius, girth, dominating number and clique number of the $\mathbb{AG}(X)$ are investigated. We have proved that $c(X)⩽\mathrm{dt}(\mathbb{AG}(X))⩽w(X)$ and $\mathrm{clique}\mathbb{AG}(X)=\chi \mathbb{AG}(X)=c(X)$.

13 pages