Quantitative Obata's theorem in discrete setting

2025/08/28 by Liu, Shiping, Zhou, Chiyu
#Combinatorics (math.CO) #Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2508.20815

Abstract

Under mild assumptions, we show that a connected weighted graph G with lower Ricci curvature bound K>0 in the sense of Bakry-Émery and the d-th non-zero Laplacian eigenvalue λ_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.

Citations

• The rigidity of eigenvalues on Kähler manifolds with positive Ricci lower bound
• Graph Similarity and Approximate Isomorphism
• Rigidity properties of the hypercube via Bakry-Emery curvature
• Curvature and transport inequalities for Markov chains in discrete spaces
• Eigenvalue Ratios of Non-Negatively Curved Graphs
• Pincement spectral en courbure de Ricci positive

Related

• Rigidity properties of the hypercube via Bakry-Emery curvature
• Ricci curvature, Bruhat graphs and Coxeter groups
• Bounding the diameter and eigenvalues of amply regular graphs via Lin-Lu-Yau curvature
• Bakry-Émery curvature on graphs as an eigenvalue problem
• Discrete quantitative nodal theorem