Bounds on curvature in regular graphs

TL;DR

This paper investigates curvature properties in regular graphs, developing techniques for calculating curvature and comparing different curvature notions, with characterizations based on subgraph structures.

Contribution

It introduces methods for computing curvature in regular graphs and compares curvature-dimension and Ollivier curvature, identifying conditions for their sign agreement.

Findings

Curvature signs usually agree when no $K_3$ or $K_{2,3}$ subgraphs are present.

Characterization of graphs with positive, zero, and negative curvature.

Identification of exceptions where curvature signs differ.

Abstract

We study the curvature-dimension inequality in regular graphs. We develop techniques for calculating the curvature of such graphs, and we give characterizations of classes of graphs with positive, zero, and negative curvature. Our main result is to compare the curvature-dimension inequality in these classes to the so-called Ollivier curvature. A consequence of our results is that in the case that the graph contains no subgraph isomorphic to either $K_3$ or $K_{2,3}$ these curvatures usually have the same sign, and we characterize the exceptions.