Ricci-flat graphs with girth at least five

2013/01/01 by Yong Lin, Linyuan Lu, Lin, Yong +5 · 3 citations
Mathematics · Physics and Astronomy · #05C80 #05C81 #05C99 #Advanced Differential Geometry Research #Combinatorics (math.CO) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.CO #msc:05C80 #msc:05C81 #msc:05C99

paper · pdf · doi:10.48550/arxiv.1301.0102

14 pages, 15 figures

arxiv created 2013/01/01 · openalex publication_date 2013/01/01 · arxiv updated 2013/01/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A graph is called Ricci-flat if its Ricci-curvatures vanish on all edges. Here we use the definition of Ricci-cruvature on graphs given in [Lin-Lu-Yau, Tohoku Math., 2011], which is a variation of [Ollivier, J. Funct. Math., 2009]. In this paper, we classified all Ricci-flat connected graphs with girth at least five: they are the infinite path, cycle C_n (n≥ 6), the dodecahedral graph, the Petersen graph, and the half-dodecahedral graph. We also construct many Ricci-flat graphs with girth 3 or 4 by using the root systems of simple Lie algebras.

Citations

• Ricci curvature for metric-measure spaces via optimal transport
• Ricci curvature of Markov chains on metric spaces

Cited by

• Estimates of the first Dirichlet eigenvalue of graphs
• Halin graphs with positive Lin-Lu-Yau curvature
• A brief review on geometry and spectrum of graphs
• Ricci-flat graphs with maximum degree at most 4

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