A Note on Heat Kernel Estimates on Weighted Graphs with Two-Sided Bounds on the Weights ∗†

We reconsider estimates for the heat kernel on weighted graphs recently found by Metzger and Stollmann. In the case that the weights satisfy a positive lower bound as well as a finite upper bound, we obtain a specialized lower estimate and a proper generalization of a previous upper estimate. Estimates for the heat kernels on graphs have recently attracted the interest of mathematical physicists [1, 2, 5]. In [6], Metzger and Stollmann derived physically significant upper and lower bounds for the heat kernel on a weighted graph which also give rise to a probabilistic interpretation in terms of stochastic processes. They considered the case that the ‘heat conductance’, that is, the weight function on the edges of the graph, is bounded from above. Here, we follow their lucent method of proof under the additional assumption that there is also a strictly positive lower bound on the weights. This case seems to be of physical relevance, for example, in the modeling of heat transport in structured media. It turns out that, while the lower estimate takes on a special form under this assumption, the upper estimate becomes properly generalize