On Lichnerowicz sharp distance-regular graphs

Abstract

The first non-zero Laplacian eigenvalue $\hat{I}{»}_1$ of a finite graph is bounded below by its minimum Lin--Lu--Yau curvature $\hat{I}º$. This is a discrete analogue of the classical Lichnerowicz Theorem. A graph with $\hat{I}{»}_1=\hat{I}º$ is called Lichnerowicz sharp. In this note, we give a new proof of the classification of Lichnerowicz sharp distance-regular graphs, which was first obtained by Münch and strengthens the corresponding classification by Cushing, Kamtue, Koolen, Liu, Münch, and Peyerimhoff, 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 in its own right.

19 pages, 2 figures. We learned that this classification had already been obtained by Florentin Münch in arXiv:2205.15857, where he classified a broader class of graphs, namely effective Bonnet--Myers sharp graphs. We sincerely thank Florentin Münch and one of the referees for pointing out this overlap. We note that the proof presented here is entirely different from Florentin Münch's