On Lichnerowicz sharp distance-regular graphs

TL;DR

This paper classifies all Lichnerowicz sharp distance-regular graphs, strengthening previous results by removing extra spectral conditions and providing new classifications of related graphs with positive curvature.

Contribution

It offers a complete classification of Lichnerowicz sharp distance-regular graphs and classifies amply regular Terwilliger graphs with positive curvature, advancing understanding in spectral graph theory.

Findings

Complete classification of Lichnerowicz sharp distance-regular graphs

Classification of amply regular Terwilliger graphs with positive curvature

Strengthening of previous spectral classification results

Abstract

The first non-zero Laplacian eigenvalue $\lambda_1$ of a finite graph is bounded below by its minimum Lin--Lu--Yau curvature $\kappa$. This is a discrete analogue of the classical Lichnerowicz Theorem. A graph with $\lambda_1=\kappa$ is called Lichnerowicz sharp. In this note, we completely classify all Lichnerowicz sharp distance-regular graphs. Our result substantially strengthens the corresponding classification by Cushing, Kamtue, Koolen, Liu, M\"unch, and Peyerimhoff (Adv. Math. 2020), which required an extra spectral condition. As a key preparatory step, we provide a classification of all amply regular Terwilliger graphs with positive Lin-Lu-Yau curvature, a result that is interesting of its own right.