On Lichnerowicz sharp distance-regular graphs
Abstract
The first non-zero Laplacian eigenvalue of a finite graph is bounded below by its minimum Lin--Lu--Yau curvature . This is a discrete analogue of the classical Lichnerowicz Theorem. A graph with 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.
Cite
@article{arxiv.2602.10396,
title = {On Lichnerowicz sharp distance-regular graphs},
author = {Kaizhe Chen and Shiping Liu and Heng Zhang},
journal= {arXiv preprint arXiv:2602.10396},
year = {2026}
}
Comments
18 pages, 2 figures. This preprint builds on Section 4 of the arXiv paper arXiv:2409.06418v2 (arXiv:2409.06418v2). We extend the classification result presented there to obtain a complete classification of Lichnerowicz sharp distance-regular graphs