Asian Journal of Mathematics

Volume 25 (2021)

Number 6

Ricci-flat graphs with maximum degree at most $4$

Pages: 757 – 814



Shuliang Bai (Shing-Tung Yau Center, Southeast University, Nanjing, China)

Linyuan Lu (University of South Carolina, Columbia, S.C., U.S.A.)

Shing-Tung Yau (Harvard University, Cambridge, Massachusetts, U.S.A.)


A graph is called Ricci-flat if its Ricci curvatures vanish on all edges, here the definition of Ricci curvature on graphs was given by Lin–Lu–Yau [7]. The authors in [8] and [3] obtained a complete characterization for all Ricci-flat graphs with girth at least five. In this paper, we completely determined all Ricci-flat graphs with maximum degree at most $4$.


Ricci curvature, Ricci-flat graph, graph construction

2010 Mathematics Subject Classification


The second-named author was supported in part by NSF grant DMS 1600811 and ONR grant N00014-17-1-2842.

The third-named author was supported by NSF grant DMS-1607871: Analysis, Geometry and Mathematical Physics; and by NSF grant DMS-1418252, Collaborative Research: Geometric Analysis for Computer and Social Networks.

Received 28 November 2018

Accepted 25 June 2021

Published 24 October 2022