Bounding the diameter and eigenvalues of amply regular graphs via Lin-Lu-Yau curvature

TL;DR

This paper establishes sharp bounds on the diameter and eigenvalues of amply regular graphs by analyzing their Lin-Lu-Yau curvature, linking discrete Ricci curvature with local graph structures.

Contribution

It introduces new bounds for amply regular graphs' curvature, diameter, and eigenvalues, using novel methods connecting Ricci curvature and local matching properties.

Findings

Sharp lower bounds for Lin-Lu-Yau curvature in amply regular graphs.

New diameter bounds derived from curvature estimates.

Eigenvalue bounds established based on local structure analysis.

Abstract

An amply regular graph is a regular graph such that any two adjacent vertices have $\alpha$ common neighbors and any two vertices with distance $2$ have $\beta$ common neighbors. We prove a sharp lower bound estimate for the Lin--Lu--Yau curvature of any amply regular graph with girth $3$ and $\beta>\alpha$. The proof involves new ideas relating discrete Ricci curvature with local matching properties: This includes a novel construction of a regular bipartite graph from the local structure and related distance estimates. As a consequence, we obtain sharp diameter and eigenvalue bounds for amply regular graphs.