Recent progress has shown exciting links between the spectral types of infinite graphs and the eigenvector behavior of finite approximations of these graphs. While it seems very intuitive that absolutely continuous spectrum should be linked to delocalized behavior of eigenvectors, this has so far only been proven rigorously for special families of graphs: trees and periodic crystals. It should be expected that such a relation holds more generally, perhaps up to accepting a weaker notion of delocalization. On the other hand, from a quantum chaos point of view, negative curvature implies delocalized behavior for eigenvectors. There are several notions of curvature for graphs, and it would be interesting to explore if there are similar implications in this context. This research in pairs will focus extensively on all these questions.