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.
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