Consider a countable set and a weighted, uniformly locally finite graph. In his paper "Parabolic Harnack inequality and estimates of Markov chains on graphs", Delmotte proves that parabolic Harnack inequalities and discrete heat kernel estimates each are equivalent to certain geometric properties of the graph. These include the volume doubling property, a Poincaré inequality and the so-called "delta condition" which ensures that edge weights are not arbitrarily small compared to the degrees of their incident vertices.
In the main part of the talk, this result and a slight modification being found in a book by Barlow will be discussed. Furthermore, some thoughts on the question will be presented, how Delmotte's main result could be modified if one wants to omit the delta condition.