Some properties of the zero-set intersection graph of C(X) and its line graph

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 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 Neumann regular ring if and only if C(X) is an almost regular ring and for all f ∈ V (Γ(C(X))) there exists g ∈ V (Γ(C(X))) such that Z(f) ∩ Z(g) = ∅ and {f, g} dominates Γ(C(X)). Finally, we derive some properties of the line graph
of Γ(C(X)).