A brief review on geometry and spectrum of graphs

2012/04/14 by Yong Lin, Shing-Tung Yau, Shing–Tung Yau +2
Mathematics · Medicine · Physics and Astronomy · #Advanced Differential Geometry Research #Combinatorics (math.CO) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Ophthalmology and Eye Disorders #Spectral Theory (math.SP) #math.CO #math.DG #math.SP

paper · pdf · doi:10.48550/arxiv.1204.3168

24 pages, 6 figures

arxiv created 2012/04/14 · openalex publication_date 2012/04/14 · arxiv updated 2012/04/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This is a survey paper. We study the Ricci curvature and spectrum of graphs, as well as the exterior forms and deRahm cohomology on graphs.

Citations

• Ricci-flat graphs with girth at least five
• Ollivier’s Ricci Curvature, Local Clustering and Curvature-Dimension Inequalities on Graphs
• Ricci curvature of Markov chains on metric spaces

Related

• Curvature, geometry and spectral properties of planar graphs
• Geometric and spectral properties of directed graphs under a lower Ricci curvature bound
• Ricci curvature and eigenvalue estimate on locally finite graphs
• Ricci curvature, Bruhat graphs and Coxeter groups
• Combinatorial, Bakry-Émery, Ollivier's Ricci curvature notions and their motivation from Riemannian geometry