# Discrete versions of the Li-Yau gradient estimate

## TL;DR

This paper extends the Li-Yau gradient estimate to discrete graph settings, providing new inequalities, sharpness results, and computational tools for analyzing heat equations on graphs.

## Contribution

It introduces discrete variants of the Li-Yau gradient estimate, establishes new computation rules, and defines a relaxation function for the differential Harnack inequality on graphs.

## Key findings

- Variants of Li-Yau gradient estimate proved for graphs
- Sharpness of estimates demonstrated on certain graphs
- New computational rules for discrete differential operators

## Abstract

We study positive solutions to the heat equation on graphs. We prove variants of the Li-Yau gradient estimate and the differential Harnack inequality. For some graphs, we can show the estimates to be sharp. We establish new computation rules for differential operators on discrete spaces and introduce a relaxation function that governs the time dependency in the differential Harnack estimate.