  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 $\Gamma
(C(X))$.
