Optimal spectral rigidity of the hypercube via Bakry--Émery curvature

Abstract

Hypercube graphs are fundamental model spaces of positive curvature in discrete comparison geometry. We establish the following spectral rigidity theorem. Let $G$ be a finite, connected, simple, unweighted graph with Bakry--Émery curvature bounded below by $K>0$. Denote by $Δ$ the maximum degree of $G$, and let $0=λ_0<λ_1\leq\cdots$ be the eigenvalues of the non-normalized Laplacian. Then $$ λ_{Δ-1}=K \quad\Longrightarrow\quad G\cong H_Δ, $$ where $H_Δ$ is the $Δ$-dimensional hypercube graph. Thus, in the unweighted setting, the multiplicity condition $λ_Δ=K$ appearing in the hypercube rigidity theorem of Liu, Münch, and Peyerimhoff can be weakened to $λ_{Δ-1}=K$. This improvement is optimal. The restriction to unweighted graphs is essential: the strengthened rigidity statement fails in the weighted setting. Our argument is built upon an interplay between the global spectral embedding induced by the first eigenspace and a local analysis of curvature matrices.