On Lichnerowicz sharp distance-regular graphs

The first non-zero Laplacian eigenvalue $λ_1$ of a finite graph is bounded below by its minimum Lin--Lu--Yau curvature $κ$. This is a discrete analogue of the classical Lichnerowicz Theorem. A graph with $λ_1=κ$ 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.