Spectral structure of the Laplacian on a covering graph

There are a lot of researches on the spectrum of the discrete Laplacian on an infinite graph in various areas. One of main topics among them is to characterize the spectral structure in terms of a certain geometric property of the graph. We focus on the possibility of the absence of eigenvalues and give a criterion for this in terms of combinatorial property of a graph. A graph G is a connected, locally finite graph and A(G) is the set of its oriented edges. For an edge e ∈ A(G), the origin vertex and the terminal one of e are denoted by o(e) and t(e), respectively. We define the discrete Laplacian by