On Lichnerowicz sharp distance-regular graphs
Kaizhe Chen, Shiping Liu, Heng Zhang
TL;DR
This work addresses the discrete analogue of the Lichnerowicz rigidity problem in graph theory by classifying all distance-regular graphs that attain equality in the Lin–Lu–Yau curvature bound, i.e., graphs with $λ_1 = \min_{xy\in E} κ(x,y)$. The authors develop a two-pronged approach: first classify amply regular Terwilliger graphs with positive $κ$ via curvature bounds and structural reductions, then leverage spectral relations for line graphs to obtain a complete list of Lichnerowicz sharp distance-regular graphs. The resulting catalogue comprises $CP(n)$, $H(d,n)$, $J(n,k)$, $Q^n_{(2)}$, the Schläfli graph, and the Gosset graph, and it shows that a prior spectral condition is unnecessary. Overall, the paper advances discrete Obata-type rigidity in Lin–Lu–Yau geometry and delivers a precise, highly symmetric graph catalog with extremal spectral-curvature properties.
Abstract
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 completely classify all Lichnerowicz sharp distance-regular graphs. Our result substantially strengthens the corresponding classification by Cushing, Kamtue, Koolen, Liu, Münch, 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.