Integral Ricci Curvature for Graphs

TL;DR

This paper introduces an integral Ricci curvature concept for graphs, providing new geometric estimates like diameter bounds and eigenvalue bounds that generalize previous results without requiring positive curvature.

Contribution

It defines integral Ricci curvature for graphs and derives several fundamental geometric and spectral estimates that extend prior work to graphs with non-positive curvature.

Findings

Proves a Bonnet-Myers-type diameter estimate.

Establishes a Moore-type vertex count estimate.

Derives a Lichnerowicz-type eigenvalue bound.

Abstract

We introduce the notion of integral Ricci curvature $I_{\kappa_0}$ for graphs, which measures the amount of Ricci curvature below a given threshold $\kappa_0$. We focus our attention on the Lin-Lu-Yau Ricci curvature. As applications, we prove a Bonnet-Myers-type diameter estimate, a Moore-type estimate on the number of vertices of a graph in terms of the maximum degree $d_M$ and diameter $D$, and a Lichnerowicz-type estimate for the first eigenvalue $\lambda_1$ of the Graph Laplacian, generalizing the results obtained by Lin, Lu, and Yau. All estimates are uniform, depending only on geometric parameters like $\kappa_0$, $I_{\kappa_0}$, $d_M$, or $D$, and do not require the graphs to be positively curved.