Remarks on Li-Yau inequality on graphs

TL;DR

This paper investigates Li-Yau gradient estimates for heat equations on graphs under curvature conditions, deriving Harnack inequalities, heat kernel estimates, and Hamilton gradient estimates to advance understanding of heat flow in discrete structures.

Contribution

It introduces new Li-Yau gradient estimates on graphs under the $CD(n,-K)$ curvature condition, extending classical results to discrete settings.

Findings

Derived Harnack inequalities for heat equations on graphs.

Established heat kernel estimates under curvature conditions.

Presented a type of Hamilton gradient estimates for graph heat equations.

Abstract

In this paper, we study Li-Yau gradient estimates for the solutions $u$ to the heat equation $\partial_tu=\Delta u$ on graphs under the curvature condition $CD(n,-K)$ introduced by Bauer et al. in \cite{BHLLMY}. As applications, we derive Harnack inequalities and heat kernel estimates on graphs. Also we present a type of Hamilton gradient estimates.