Year: 2026
Type: preprint
Abstract: 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, after removing a canonical collection of edges on which every $K$-eigenfunction is constant, the resulting graph has a canonical bundle structure. Its fibers are regular, have similar structure with hypercubes, and are Laplacian-cospectral with hypercubes, although they need not themselves be hypercubes. If the base graph is nontrivial, then it satisfies $\mathrm{CD}(K,\infty)$ and has first eigenvalue strictly greater than $K$. As a consequence, if the vertex degree in $G$ is constant along each canonical fiber, then every fiber is a hypercube and $G$ is a hypercube bundle. Conversely, for every $d\geq 4$, we construct Lichnerowicz-sharp graphs with non-hypercube canonical fibers of degree $d$.
Source: arXiv (Cornell University)
Authors: Ding, Yanlong, Liu, Shiping, Zhou, Chiyu
Institution: University of Science and Technology of China
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G7920131011
OpenAlex ID: W7204226322
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