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