Ricci-flat cubic graphs with girth five
David Cushing, Riikka Kangaslampi, Yong Lin, Shiping Liu, Linyuan Lu, and Shing-Tung Yau
TL;DR
This paper classifies all connected, simple, 3-regular graphs with girth at least 5 that are Ricci-flat, identifying only the Petersen graph, Triplex, and dodecahedral graph as such, correcting previous classifications.
Contribution
It provides a complete classification of Ricci-flat cubic graphs with girth at least 5, including the previously omitted Triplex.
Findings
Only Petersen, Triplex, and dodecahedral graphs are Ricci-flat with girth ≥ 5.
Corrects previous classification by including the Triplex.
Uses Lin-Lu-Yau's Ricci curvature definition on graphs.
Abstract
We classify all connected, simple, 3-regular graphs with girth at least 5 that are Ricci-flat. We use the definition of Ricci curvature on graphs given in Lin-Lu-Yau, Tohoku Math., 2011, which is a variation of Ollivier, J. Funct. Anal., 2009. A graph is Ricci-flat, if it has vanishing Ricci curvature on all edges. We show, that the only Ricci-flat cubic graphs with girth at least 5 are the Petersen graph, the Triplex and the dodecahedral graph. This will correct the classification in Lin-Lu-Yau, Comm. Anal. Geom., 2014, that misses the Triplex.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Ophthalmology and Eye Disorders · Advanced Differential Geometry Research