In this paper, we introduce a graph structure, called zero-set intersection graph Γ(C(X)), on the ring of real valued continuous functions, C(X), on a Tychonoff space X. We show that the graph is connected and triangulated. We also study the interrelationship of cliques of Γ(C(X)) and ideals in C(X) which helps to characterize the structure of maximal cliques of Γ(C(X)) by different kind of maximal ideals of C(X). We show that there are at least 2 c many different maximal cliques which are never graph isomorphic to each other. Furthermore, we study the neighbourhood properties of a vertex and show its connection with the topology of X and algebraic properties of C(X). Finally, it is shown that two graphs are isomorphic if and only if the corresponding rings are isomorphic if and only if the corresponding topologies are homeomorphic either for first countable topological spaces or for realcompact topological spaces.