# Quantitative Obata's theorem in discrete setting

## TL;DR

This paper proves that certain weighted graphs with positive Ricci curvature and eigenvalues close to the curvature bound resemble hypercube graphs both structurally and in eigenfunction behavior, extending Obata's theorem to discrete settings.

## Contribution

It establishes a quantitative discrete analogue of Obata's theorem, showing that graphs with specific spectral and curvature conditions are close to hypercube graphs.

## Key findings

- Graphs with Ricci curvature > 0 and eigenvalues near the bound are structurally close to hypercubes.
- Such graphs are close to hypercube graphs in Frobenius distance.
- Eigenfunctions of these graphs resemble those of hypercube graphs.

## Abstract

Under mild assumptions, we show that a connected weighted graph $G$ with lower Ricci curvature bound $K>0$ in the sense of Bakry-\'Emery and the $d$-th non-zero Laplacian eigenvalue $\lambda_d$ close to $K$, with $d$ being the maximal combinatorial vertex degree of $G$, has an underlying combinatorial structure of the $d$-dimensional hypercube graph. Moreover, such a graph $G$ is close in terms of Frobenius distance to a properly weighted hypercube graph. Furthermore, we establish their closeness in terms of eigenfunctions. Our results can be viewed as discrete analogies of the almost rigidity theorem and quantitative Obata's theorem on Rimennian manifolds.