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) \leqslant \mathrm{dt}(\mathbb{AG}(X)) \leqslant w(X) $ and $ \mathrm{clique} \mathbb{AG}(X) = \chi \mathbb{AG}(X) = c(X) $.