# Non-negative Ollivier curvature on graphs, reverse Poincar\'e inequality, Buser inequality, Liouville property, Harnack inequality and eigenvalue estimates

## TL;DR

This paper establishes a new inequality for graphs with non-negative Ollivier curvature, leading to several geometric and spectral results such as Buser inequality, Liouville property, and eigenvalue bounds.

## Contribution

It introduces a novel heat semigroup inequality equivalent to reverse Poincaré inequality for such graphs, enabling new geometric and spectral insights.

## Key findings

- Proves a Wasserstein distance decay estimate for heat semigroup on graphs
- Derives Buser inequality and Liouville property from the curvature condition
- Provides a lower bound for the first eigenvalue in terms of graph diameter

## Abstract

We prove that for combinatorial graphs with non-negative Ollivier curvature, one has \[ \|P_t \mu - P_t \nu\|_1 \leq \frac{W_1(\mu,\nu)}{\sqrt{t}} \] for all probability measures $\mu,\nu$ where $P_t$ is the heat semigroup and $W_1$ is the $\ell_1$-Wasserstein distance. This turns out to be an equivalent formulation of a version of reverse Poincar\'e inequality. Furthermore, this estimate allows us to prove Buser inequality, Liouville property and the the eigenvalue estimate $\lambda_1 \geq \log(2)/\operatorname{diam}^2$.