Davies–Gaffney–Grigor’yan Lemma on graphs

We prove a variant of the Davies-Gaffney-Grigor'yan Lemma for the continuous time heat kernel on graphs. We use it together with the Li-Yau inequality, to obtain strong heat kernel estimates for graphs satisfying the exponential curvature dimension inequality. 1 Introduction and main results 1032 1.1 Introduction 1032 1.2 Main results and organization of the paper 1033 1.3 Setting 1035 2 The Li-Yau inequality on graphs 1036 2.1 The exponential curvature dimension inequality 1037 2.2 Gradient estimates and the Harnack inequality 1038 2.3 Applications of the Li-Yau inequality 1041 3 Davies-Gaffney-Grigor'yan Lemma 1043