Li-Yau inequality under $CD(0,n)$ on graphs

TL;DR

This paper establishes a Li-Yau inequality for graphs under the $CD(0,n)$ curvature condition, introduces a modified heat equation, and derives volume doubling and non-existence of certain expander graphs, advancing discrete Ricci curvature theory.

Contribution

It introduces a new non-linear heat equation approach and proves the Li-Yau inequality under $CD(0,n)$ on graphs, solving a major open problem in discrete Ricci curvature.

Findings

Proves exponential decay of $ abla u$ under $CD(K, fty)$

Establishes Li-Yau inequality $- riangle u_t \\leq \frac{n}{2t}$ under $CD(0,n)$

Deduces volume doubling property and non-existence of certain expander graphs

Abstract

We introduce a modified non-linear heat equation $\partial_t u = \Delta u + \Gamma u$ as a substitute of $\log P_t f$ where $P_t$ is the heat semigroup. We prove an exponential decay of $\Gamma u$ under the Bakry Emery curvature condition $CD(K,\infty)$ and prove the Li-Yau inequality $-\Delta u_t \leq \frac{n}{2t}$ under the Bakry Emery curvature condition $CD(0,n)$. From this, we deduce the volume doubling property which solves a major open problem in discrete Ricci curvature. As an application, we show that there exist no expander graphs satisfying $CD(0,n)$.