Spectrally positive Bakry-\'Emery Ricci curvature on graphs

TL;DR

This paper explores the effects of non-constant Ricci curvature bounds on graphs, establishing key geometric and analytic inequalities under positive and small negative curvature conditions.

Contribution

It introduces new bounds and inequalities for graphs with spectrally positive Bakry-Émery Ricci curvature, including eigenvalue estimates and diameter bounds, allowing for some negative curvature.

Findings

Proves a Lichnerowicz eigenvalue estimate.

Establishes diameter bounds and Harnack inequalities.

Shows finiteness of the fundamental group under certain curvature conditions.

Abstract

We investigate analytic and geometric implications of non-constant Ricci curvature bounds. We prove a Lichnerowicz eigenvalue estimate and finiteness of the fundamental group assuming that $L+2 Ric$ is a positive operator where $L$ is the graph Laplacian. Assuming that the negative part of the Ricci curvature is small in Kato sense, we prove diameter bounds, elliptic Harnack inequality and Buser inequality. This article seems to be the first one establishing these results while allowing for some negative curvature.