  In this article, we introduce the annihilator graph of the ring
$C_\mathscr{P}(X)$, denoted by $AG(C_\mathscr{P}(X))$ and observe the effect of
the underlying Tychonoff space $X$ on various graph properties of
$AG(C_\mathscr{P}(X))$. $AG(C_\mathscr{P}(X))$, in general, lies between the
zero divisor graph and weakly zero divisor graph of $C_\mathscr{P}(X)$ and it
is proved that these three graphs coincide if and only if the cardinality of
the set of all $\mathscr{P}$-points, $X_\mathscr{P}$ is $\leq 2$. Identifying a
suitable induced subgraph of $AG(C_\mathscr{P}(X))$, called
$G(C_\mathscr{P}(X))$, we establish that both $AG(C_\mathscr{P}(X))$ and
$G(C_\mathscr{P}(X))$ share similar graph theoretic properties and have the
same values for the parameters, e.g., diameter, eccentricity, girth, radius,
chromatic number and clique number. By choosing the ring $C_\mathscr{P}(X)$
where $\mathscr{P}$ is the ideal of all finite subsets of $X$ such that
$X_\mathscr{P}$ is finite, we formulate an algorithm for coloring the vertices
of $G(C_\mathscr{P}(X))$ and thereby get the chromatic number of
$AG(C_\mathscr{P}(X))$. This exhibits an instance of coloring infinite graphs
by just a finite number of colors. We show that any graph isomorphism $\psi :
AG(C_\mathscr{P}(X)) \rightarrow AG(C_\mathscr{Q}(Y))$ maps
$G(C_\mathscr{P}(X))$ isomorphically onto $G(C_\mathscr{Q}(Y))$ as a graph and
a graph isomorphism $\phi : G(C_\mathscr{P}(X)) \rightarrow
G(C_\mathscr{Q}(Y))$ can be extended to a graph isomorphism $\psi :
AG(C_\mathscr{P}(X)) \rightarrow AG(C_\mathscr{Q}(Y))$ under a mild restriction
on the function $\phi$. Finally, we show that atleast for the rings
$C_\mathscr{P}(X)$ with finitely many $\mathscr{P}$-points, so far as the graph
properties are concerned, the induced subgraph $G(C_\mathscr{P}(X))$ is a good
substitute for $AG(C_\mathscr{P}(X)$.
