Some properties of the Zero-set Intersection graph of $C(X)$ and its Line graph

TL;DR

This paper explores the properties of the zero-set intersection graph of continuous functions on a topological space and its line graph, revealing their connectivity, girth, triangulation, domination number, and algebraic implications.

Contribution

It provides new insights into the structure and properties of zero-set intersection graphs of $C(X)$ and their line graphs, including connectivity, girth, triangulation, and algebraic characterizations.

Findings

Graphs are connected with diameter and radius 2.

Girth of the graphs is 3, and they are triangulated and hypertriangulated.

Characterization of $C(X)$ as a von Neumann regular ring.

Abstract

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 $\Gamma (C(X))$ 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 $f \in V(\Gamma (C(X)))$ there exists $g \in V(\Gamma(C(X)))$ such that $Z(f) \cap Z(g) = \phi$ and $\{f, g\}$ dominates $\Gamma(C(X))$. Finally, we derive some properties of the line graph of $\Gamma (C(X))$.