We prove that a two-sided sub-Gaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay.
Alexander Grigor'yan. Andras Telcs. "Sub-Gaussian estimates of heat kernels on infinite graphs." Duke Math. J. 109 (3) 451 - 510, 15 September 2001. https://doi.org/10.1215/S0012-7094-01-10932-0