Heat kernel and essential spectrum of infinite graphs

We study the existence and uniqueness of the heat kernel on infinite, locally finite, connected graphs.For general graphs, a uniqueness criterion, shown to be optimal, is given in terms of the maximal valence on spheres about a fixed vertex.A sufficient condition for non-uniqueness is also presented.Furthermore, we give a lower bound on the bottom of the spectrum of the discrete Laplacian and use this bound to give a condition ensuring that the essential spectrum of the Laplacian is empty.