Li-Yau inequality on graphs

TL;DR

This paper establishes a Li-Yau gradient estimate for heat kernels on graphs using a new local curvature notion, leading to insights on volume growth, Harnack inequalities, and spectral properties.

Contribution

It introduces a novel local curvature concept for graphs and proves the Li-Yau inequality under this framework, connecting curvature to heat kernel behavior.

Findings

Curvature behaves more naturally on lattices and trees.

Graphs with non-negative curvature exhibit polynomial volume growth.

Derived Harnack inequalities and heat kernel bounds.

Abstract

We prove the Li-Yau gradient estimate for the heat kernel on graphs. The only assumption 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 this curvature for lattices and trees and conclude that it behaves more naturally than the already 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 estimate, and show how it can be used to strengthen the classical Buser inequality relating the spectral gap and the Cheeger constant of a graph.