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

Heat kernel and essential spectrum of infinite graphs