Discrete curvature and abelian groups

TL;DR

This paper introduces a discrete curvature concept for graphs, especially abelian groups, and establishes inequalities linking spectral and isoperimetric properties, providing new insights into graph geometry and spectral bounds.

Contribution

It develops a computable discrete curvature notion for graphs, derives Buser-type inequalities, and bounds the Cheeger constant for abelian Cayley graphs.

Findings

Discrete curvature can be effectively computed for specific graphs.

Buser-type inequalities relate spectral gap and isoperimetric constants.

Cheeger constant bounds are tight for abelian Cayley graphs.

Abstract

We study a natural discrete Bochner-type inequality on graphs, and explore its merit as a notion of curvature in discrete spaces. An appealing feature of this discrete version seems to be that it is fairly straightforward to compute this notion of curvature parameter for several specific graphs of interest - particularly, abelian groups, slices of the hypercube, and the symmetric group under various sets of generators. We further develop this notion by deriving Buser-type inequalities (a la Ledoux), relating functional and isoperimetric constants associated with a graph. Our derivations provide a tight bound on the Cheeger constant (i.e., the edge-isoperimetric constant) in terms of the spectral gap, for graphs with nonnegative curvature, particularly, the class of abelian Cayley graphs - a result of independent interest.