Heat kernel and essential spectrum of infinite graphs Radoslaw Wojciechowski 39A1258J35heat equationheat kernelstochastic completenessgraphsbottom of the spectrumessential spectrum 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. Indiana University Mathematics Journal 2009 text pdf 10.1512/iumj.2009.58.3575 10.1512/iumj.2009.58.3575 en Indiana Univ. Math. J. 58 (2009) 1419 - 1442 state-of-the-art mathematics http://iumj.org/access/