  For any multi-graph $G$ with edge weights and vertex potential, and its
universal covering tree $\mathcal{T}$, we completely characterize the point
spectrum of operators $A_{\mathcal{T}}$ on $\mathcal{T}$ arising as pull-backs
of local, self-adjoint operators $A_{G}$ on $G$. This builds on work of Aomoto,
and includes an alternative proof of the necessary condition for point spectrum
he derived in (Aomoto, 1991). Our result gives a finite time algorithm to
compute the point spectrum of $A_{\mathcal{T}}$ from the graph $G$, and
additionally allows us to show that this point spectrum is contained in the
spectrum of $A_{G}$. Finally, we prove that typical pull-back operators have a
spectral delocalization property: the set of edge weight and vertex potential
parameters of $A_{G}$ giving rise to $A_{\mathcal{T}}$ with purely absolutely
continuous spectrum is open and its complement has large codimension.
