Diameter bounds for distance-regular graphs via long-scale Ollivier Ricci curvature

TL;DR

This paper establishes new sharp diameter bounds for distance-regular graphs by relating their diameter to long-scale Ollivier Ricci curvature, improving upon classical results and addressing longstanding questions.

Contribution

It introduces a novel approach linking diameter bounds to Ollivier Ricci curvature, enhancing existing bounds for specific classes of regular graphs.

Findings

Derived sharper diameter bounds for distance-regular graphs.

Improved diameter bounds for amply regular graphs.

Enhanced bounds for (s,c,a,k)-graphs.

Abstract

In this paper, we derive new sharp diameter bounds for distance regular graphs, which better answer a problem raised by Neumaier and Penji\' c in many cases. Our proof is built upon a relation between the diameter and long-scale Ollivier Ricci curvature of a graph, which can be considered as an improvement of the discrete Bonnet-Myers theorem. Our method further leads to significant improvements of existing diameter bounds for amply regular graphs and $(s,c,a,k)$-graphs.