  Consider a random Schr\"odinger-type operator of the form $H:=-H_X+V+\xi$
acting on a general graph $\mathscr G=(\mathscr V,\mathscr E)$, where $H_X$ is
the generator of a Markov process $X$ on $\mathscr G$, $V$ is a deterministic
potential with sufficient growth (so that $H$ has a purely discrete spectrum),
and $\xi$ is a random noise with at-most-exponential tails. We prove that $H$'s
eigenvalue point process is number rigid in the sense of Ghosh and Peres (Duke
Math. J. 166 (2017), no. 10, 1789--1858); that is, the number of eigenvalues in
any bounded domain $B\subset\mathbb C$ is determined by the configuration of
eigenvalues outside of $B$. Our general setting allows to treat cases where $X$
could be non-symmetric (hence $H$ is non-self-adjoint) and $\xi$ has long-range
dependence. Our strategy of proof consists of controlling the variance of the
trace of the semigroup $\mathrm e^{-t H}$ using the Feynman-Kac formula.
