In this article we introduce the zero-divisor graphs Γ P (X) and Γ P ∞ (X) of the two rings C P (X) and C P ∞ (X); here P is an ideal of closed sets in X and C P (X) is the aggregate of those functions in C(X), whose support lie on P. C P ∞ (X) is the P analogue of the ring C ∞ (X). We determine when the weakly zero-divisor graph WΓ P (X) of C P (X) coincides with Γ P (X). We find out conditions on the topology on X, under-which Γ P (X) (respectively, Γ P ∞ (X)) becomes triangulated/ hypertriangulated. We realize that Γ P (X) (respectively, Γ P ∞ (X)) is a complemented graph if and only if the space of minimal prime ideals in C P (X) (respectively Γ P ∞ (X)) is compact. This places a special case of this result with the choice P ≡ the ideals of closed sets in X, obtained by Azarpanah and Motamedi in [8] on a wider setting. We also give an example of a non-locally finite graph having finite chromatic number. Finally it is established with some special choices of the ideals P and Q on X and Y respectively that the rings C P (X) and C Q (Y) are isomorphic if and only if Γ P (X) and Γ Q (Y) are isomorphic.