Volume growth, curvature, and Buser-type inequalities in graphs

TL;DR

This paper investigates how volume growth and discrete curvature influence spectral properties of graphs, providing bounds on eigenvalues and Cheeger constants, and extending Buser's inequality to discrete settings.

Contribution

It introduces new bounds on volume growth and eigenvalues in graphs based on discrete curvature, and adapts Buser's inequality to the graph context.

Findings

Volume growth bounds improve under Ollivier curvature constraints.

Eigenvalues of graph Laplacians can be bounded using volume growth and Hardy inequalities.

The paper establishes conditions for tight Cheeger inequalities and Buser's inequality in graphs.

Abstract

We study the volume growth of metric balls as a function of the radius in discrete spaces, and focus on the relationship between volume growth and discrete curvature. We improve volume growth bounds under a lower bound on the so-called Ollivier curvature, and discuss similar results under other types of discrete Ricci curvature. Following recent work in the continuous setting of Riemannian manifolds (by the first author), we then bound the eigenvalues of the Laplacian of a graph under bounds on the volume growth. In particular, the spectral gap of the graph can be bounded using a weighted discrete Hardy inequality and the higher eigenvalues of the graph can be bounded by the eigenvalues of a tridiagonal matrix times a multiplicative factor, both of which only depend on the volume growth of the graph. As a direct application, we relate the eigenvalues to the Cheeger isoperimetric constant. Using these methods, we describe classes of graphs for which the Cheeger inequality is tight on the second eigenvalue. We also describe a method for proving Buser's inequality in graphs, particularly under a lower bound assumption on curvature.