Lin-Lu-Yau curvature and diameter of amply regular graphs

TL;DR

This paper investigates the Lin-Lu-Yau curvature of amply regular graphs with small girth, establishing bounds, showing positivity for conference graphs, and providing geometric diameter estimates.

Contribution

It offers new lower bounds for curvature in amply regular graphs and a geometric proof for diameter estimates, expanding understanding of graph curvature and structure.

Findings

Conference graphs have positive Lin-Lu-Yau curvature.

Derived diameter bounds for amply regular graphs with girth 3 or 4.

Provided a geometric proof of classical diameter estimates.

Abstract

By Hall's marriage theorem, we study lower bounds of the Lin-Lu-Yau curvature of amply regular graphs with girth $3$ or $4$ under different parameter restrictions. As a consequence, we show that each conference graph has positive Lin-Lu-Yau curvature. Our approach also provides a geometric proof of a classical diameter estimate for amply regular graphs in the case of girth $4$ and some special cases of girth $3$.