Li-Yau inequality on graphs

Abstract

AbstractWe prove the Li-Yau gradient estimate for the heat kernel on graphs. The onlyassumption is a variant of the curvature-dimension inequality, which is purely local,and can be considered as a new notion of curvature for graphs. We compute thiscurvature for lattices and trees and conclude that it behaves more naturally than thealready existing notions of curvature. Moreover, we show that if a graph has non-negative curvature then it has polynomial volume growth.We also derive Harnack inequalities and heat kernel bounds from the gradient esti-mate, and show how it can be used to strengthen the classical Buser inequality relatingthe spectral gap and the Cheeger constant of a graph. 1 Introduction and main ideas In their celebrated work [15] Li and Yau proved an upper bound on the gradient ofpositive solutions of the heat equation. In its simplest form, for an n-dimensional com-pact manifold with non-negative Ricci curvature the Li-Yau gradient estimate statesthat a positive solution uof the heat equation (∆−∂