Heat Kernel Estimates and Parabolic Harnack Inequalities on Graphs and Resistance Forms

We summarize recent work on heat kernel estimates and parabolic Harnack inequalities for graphs, where the time scale is the β -th power of the space scale for some β ≥ 2 . We then discuss self-adjoint operators induced by resistance forms. Using a resistance metric, we give a simple condition for detailed heat kernel estimates and parabolic Harnack inequalities. As an application, we show that on trees a detailed two-sided heat kernel estimate is equivalent to some volume growth condition.