On Lichnerowicz sharp distance-regular graphs

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 give a new proof of the classification of Lichnerowicz sharp distance-regular graphs, which was first obtained by M\"unch and strengthens the corresponding classification by Cushing, Kamtue, Koolen, Liu, M\"unch, 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.