Ricci curvature, graphs and eigenvalues

TL;DR

This paper introduces a novel method to compute Ricci curvature of graphs using eigenvalues of matrices derived from local adjacency, with applications to Cayley graphs of Coxeter groups and related isoperimetric inequalities.

Contribution

It presents a new eigenvalue-based approach to determine discrete Ricci curvature and applies it to specific classes of graphs, providing bounds and inequalities.

Findings

Computed Ricci curvature for Cayley graphs of Coxeter groups.

Established bounds on Ricci curvature using eigenvalues.

Derived an isoperimetric inequality for these graphs.

Abstract

We express the discrete Ricci curvature of a graph as the minimal eigenvalue of a family of matrices, one for each vertex of a graph whose entries depend on the local adjaciency structure of the graph. Using this method we compute or bound the Ricci curvature of Cayley graphs of finite Coxeter groups and affine Weyl groups. As an application we obtain an isoperimetric inequality that holds for all Cayley graphs of finite Coxeter groups.