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Curvatures, graph products and Ricci flatness.

Cushing, David and Kamtue, Supanat and Kangaslampi, Riikka and Liu, Shiping and Peyerimhoff, Norbert (2021) 'Curvatures, graph products and Ricci flatness.', Journal of graph theory., 96 (4). pp. 522-553.


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.

Item Type:Article
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Publisher statement:© 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.
Date accepted:14 September 2020
Date deposited:27 September 2020
Date of first online publication:12 October 2020
Date first made open access:14 October 2020

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