Zero-divisor graph of the rings $C_mathscr{P}(X)$ and $C^mathscr{P}_infty(X)$

In this article we introduce the zero-divisor graphs $Gamma_mathscr{P}(X)$ and $Gamma^mathscr{P}_infty(X)$ of the two rings $C_mathscr{P}(X)$ and $C^mathscr{P}_infty(X)$; here $mathscr{P}$ is an ideal of closed sets in $X$ and $C_mathscr{P}(X)$ is the aggregate of those functions in $C(X)$, whose support lie on $mathscr{P}$. $C^mathscr{P}_infty(X)$ is the $mathscr{P}$ analogue of the ring $C_infty (X)$. We find out conditions on the topology on $X$, under-which $Gamma_mathscr{P}(X)$ (respectively, $Gamma^mathscr{P}_infty(X)$) becomes triangulated/ hypertriangulated. We realize that $Gamma_mathscr{P}(X)$ (respectively, $Gamma^mathscr{P}_infty(X)$) is a complemented graph if and only if the space of minimal prime ideals in $C_mathscr{P}(X)$ (respectively $Gamma^mathscr{P}_infty(X)$) is compact. This places a special case of this result with the choice $mathscr{P}equiv$ the ideals of closed sets in $X$, obtained by Azarpanah and Motamedi in cite{Azarpanah} 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 $mathscr{P}$ and $mathscr{Q}$ on $X$ and $Y$ respectively that the rings $C_mathscr{P}(X)$ and $C_mathscr{Q}(Y)$ are isomorphic if and only if $Gamma_mathscr{P}(X)$ and $Gamma_mathscr{Q}(Y)$ are isomorphic.