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Curvatures, Graph Products and Ricci Flatness

Cushing, David; Kamtue, Supanat; Kangaslampi, Riikka; Liu, Shiping; Peyerimhoff, Norbert

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David Cushing

Supanat Kamtue

Riikka Kangaslampi

Shiping Liu


In this paper, we compare Ollivier–Ricci curvature and Bakry–Émery curvature notions on combinatorial graphs and discuss connections to various types of Ricci flatness. We show that nonnegativity of Ollivier–Ricci curvature implies the nonnegativity of Bakry–Émery curvature under triangle‐freeness and an additional in‐degree condition. We also provide examples that both conditions of this result are necessary. We investigate relations to graph products and show that Ricci flatness is preserved under all natural products. While nonnegativity of both curvatures is preserved under Cartesian products, we show that in the case of strong products, nonnegativity of Ollivier–Ricci curvature is only preserved for horizontal and vertical edges. We also prove that all distance‐regular graphs of girth 4 attain their maximal possible curvature values.


Cushing, D., Kamtue, S., Kangaslampi, R., Liu, S., & Peyerimhoff, N. (2021). Curvatures, Graph Products and Ricci Flatness. Journal of Graph Theory, 96(4), 522-553.

Journal Article Type Article
Acceptance Date Sep 14, 2020
Online Publication Date Oct 12, 2020
Publication Date 2021-03
Deposit Date Sep 26, 2020
Publicly Available Date Oct 14, 2020
Journal Journal of Graph Theory
Print ISSN 0364-9024
Electronic ISSN 1097-0118
Publisher Wiley
Peer Reviewed Peer Reviewed
Volume 96
Issue 4
Pages 522-553


Published Journal Article (Advance online version) (1.7 Mb)

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Copyright Statement
Advance online version © 2020 The Authors. Journal of Graph Theory published by Wiley Periodicals LLC. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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