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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.02056 |
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Table of Contents:
- Let $C(X)$ be the ring of all continuous real valued functions defined on a completely regular Hausdorff topological space $X$. The zero-set intersection graph $Γ(C(X))$ of $C(X)$ is a simple graph with vertex set all non units of $C(X)$ and two vertices are adjacent if the intersection of the zero sets of the functions is non empty. In this paper, we study the zero-set intersection graph of $C(X)$ and its line graph. We show that if $X$ has more than two points, then these graphs are connected with diameter and radius 2. We show that the girth of the graph is 3 and the graphs are both triangulated and hypertriangulated. We find the domination number of these graphs and finally we prove that $C(X)$ is a von Neuman regular ring if and only if $C(X)$ is an almost regular ring and for all $f \in V(Γ(C(X)))$ there exists $g \in V(Γ(C(X)))$ such that $Z(f) \cap Z(g) = ϕ$ and $\{f, g\}$ dominates $Γ(C(X))$. Finally, we derive some properties of the line graph of $Γ(C(X))$.