Ricci curvature, graphs and eigenvalues

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.

Bakry-Émery curvature on graphs as an eigenvalue problem

“…Note: After the submission of our first arXiv version, we became aware of the work by Siconolfi [50,51] in which the ∞-Bakry-Émery curvature K G,x (∞) is also formulated as an eigenvalue problem in the special case of non-weighted graphs.…”

Ricci curvature of Bruhat orders

“…In this work we consider the Ricci curvature of the Hasse graphs of the Bruhat order of finite irreducible Coxeter systems. This is a considerably harder problem that those studied by the same author on [25] and [26] because not all vertices are 'isomorphic' in these graphs.…”

“…In [24], [25] and [26] the Ricci curvature of Bruhat graphs of Coxeter groups and of the Hasse graphs of the weak orders of Coxeter and affine Weyl groups is studied.…”

“…Our approach is to apply to the Bruhat orders some results obtained in [26] bound the Ricci curvature of a graph in terms of the degree of its vertices.…”