Rigidity of the Bonnet-Myers inequality for graphs with respect to Ollivier Ricci curvature

TL;DR

This paper classifies all self-centered Bonnet-Myers sharp graphs using Ollivier Ricci curvature, revealing their structure and connections to other curvature notions, and proposes a conjecture relating to Bakry-Émery curvature.

Contribution

It provides a complete classification of Bonnet-Myers sharp graphs in the Ollivier Ricci curvature framework and links this sharpness to other curvature concepts.

Findings

Classification of all self-centered Bonnet-Myers sharp graphs

Demonstration that Bonnet-Myers sharpness implies Lichnerowicz sharpness

Relation of Bonnet-Myers sharpness to Bakry-Émery curvature upper bounds

Abstract

We introduce the notion of Bonnet-Myers and Lichnerowicz sharpness in the Ollivier Ricci curvature sense. Our main result is a classification of all self-centered Bonnet-Myers sharp graphs (hypercubes, cocktail party graphs, even-dimensional demi-cubes, Johnson graphs $J(2n,n)$, the Gosset graph and suitable Cartesian products). We also present a purely combinatorial reformulation of this result. We show that Bonnet-Myers sharpness implies Lichnerowicz sharpness. We also relate Bonnet-Myers sharpness to an upper bound of Bakry-\'Emery $\infty$-curvature, which motivates a generalconjecture about Bakry-\'Emery $\infty$-curvature.