超立方体的最优谱刚性：基于 Bakry–Émery 曲率
Regular Lichnerowicz-sharp graphs are hypercube bundles
AI总结：
通过 Bakry–Émery 曲率下界，证明超立方体图在未加权图中的谱刚性：若最大度为Δ，则特征值λ_{Δ-1}=K蕴含图同构于Δ维超立方体，且该结果最优。
摘要：
超立方体图是离散比较几何中正曲率的基本模型空间。我们建立了以下谱刚性定理。设 $G$ 是一个有限、连通、简单、未加权的图，其 Bakry--Émery 曲率有下界 $K>0$。记 $Δ$ 为 $G$ 的最大度，并令 $0=λ_0<λ_1\leq\cdots$ 为非归一化拉普拉斯算子的特征值。那么 $$ λ_{Δ-1}=K \quad\Longrightarrow\quad G\cong H_Δ, $$ 其中 $H_Δ$ 是 $Δ$ 维超立方体图。因此，在未加权设定下，Liu、Münch 和 Peyerimhoff 的超立方体刚性定理中出现的重数条件 $λ_Δ=K$ 可以减弱为 $λ_{Δ-1}=K$。这一改进是最优的。对未加权图的限制是必要的：在加权设定下，加强的刚性陈述不成立。我们的论证建立在由第一特征空间诱导的全局谱嵌入与曲率矩阵的局部分析之间的相互作用之上。
英文摘要：
Hypercube graphs are fundamental model spaces of positive curvature in discrete comparison geometry. Let $G$ be a finite, connected, simple, unweighted graph with Bakry--Émery curvature bounded below by $K$. We call $G$ Lichnerowicz-sharp if its first non-zero non-normalized Laplacian eigenvalue $λ_1=K$. We prove that all regular Lichnerowicz-sharp graphs are hypercube bundles with constant Bakry-Émery curvature $2$. A hypercube bundle is a graph bundle whose fiber graphs are hypercubes. As applications, we show that any $d$-regular Lichnerowicz-sharp graph with the multiplicity $m_K(G)$ of $λ_1=K$ at least $d/2$ can split off a hypercube of certain dimension. Moreover, we characterize all $d$-regular Lichnerowicz-sharp graphs with $m_{K}(G)\geq d-3$. Interestingly, our result leads to the following spectral rigidity theorem of hypercubes. For a graph $G$ with maximum degree $Δ$, if the multiplicity $m_K(G)\geq Δ-1$, then $G$ is a $Δ$-dimensional hypercube. This improves, in the unweighted setting, the multiplicity condition $m_K(G)\geq Δ$ appearing in the hypercube rigidity theorem of Liu, Münch, and Peyerimhoff. This improvement is optimal.
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