arXiv 2602.103962026-02math.CO

关于利钦诺维奇尖锐距离正则图

On Lichnerowicz sharp distance-regular graphs

School of the Gifted Young, University of Science and Technology of China（中国科学技术大学少年班）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）

Kaizhe Chen, Shiping Liu, Heng Zhang

备注：: 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

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AI总结：

本文通过新证明重新分类了利钦诺维奇尖锐距离正则图，同时分类了具有正Lin-Lu-Yau曲率的 ample 正则 Terwilliger 图。

摘要：

有限图的第一个非零拉普拉斯特征值λ₁下界由其最小Lin-Lu-Yau曲率κ给出。这是经典利钦诺维奇定理的离散类比。λ₁=κ的图称为利钦诺维奇尖锐。本文给出了利钦诺维奇尖锐距离正则图分类的新证明，该分类最初由Münch获得，并加强了Cushing、Kamtue、Koolen、Liu、Münch和Peyerimhoff的相应分类，后者需要额外的谱条件。作为关键预备步骤，我们提供了所有具有正Lin-Lu-Yau曲率的amply正则Terwilliger图的分类，这一结果本身具有趣味性。

英文摘要：

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.

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