Annihilating-Ideal Graph of $C(X)$

TL;DR

This paper explores the properties of the annihilating-ideal graph of the ring of continuous functions on a topological space, linking graph, ring, and topological properties with new results on graph invariants.

Contribution

It establishes connections between topological features of space X, algebraic properties of C(X), and graph invariants of the annihilating-ideal graph, including conditions for when the graph is not triangulated.

Findings

X has an isolated point iff R is a direct summand of C(X).

The radius, girth, dominating number, and clique number of the graph are characterized.

Bounds are provided for the diameter and clique number of the graph.

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) $.