Heat kernel estimates on weighted graphs

We prove upper and lower heat kernel bounds for the Laplacian on weighted graphs which include the case that the weights have no strictly positive lower bound. Our estimates allow for a very explicit probabilistic interpretation and can be formulated in terms of a weighted metric. Interestingly, this metric is not equivalent to the intrinsic metric. The results Heat kernel estimates are a tool of fundamental importance as they usually provide a link between diierent aspects of the Laplacian: spectral theory, geometry and probability theory. In this article we study heat kernels on graphs with weighted edges. Here the geometric and probabilistic features are to a large extent contained in the properties of the weight function. Diierent aspects of such a study have attracted considerable interest, see 1, 2, 3, 4, 7]. The present work is motivated by spectral theoretic applications, namely by the aim to prove absence of absolutely continuous spectrum for Schrr odinger operators on graphs which contain enough barriers made up of edges with low conductance (low weight). To this end we had to improve partly upon estimates by Davies 5, 6] and Pang 8]. The progress refers to a sensitivity of our estimates to the following geometry which is characteristic of the application we have in mind: given two points which are separated by edges of low weight the heat kernel should "feel" this barrier of low weight edges. Another 1