Bakry–Émery curvature and diameter bounds on graphs

Liu, Shiping; Münch, Florentin; Peyerimhoff, Norbert

Abstract

We prove finiteness and diameter bounds for graphs having a positive Ricci-curvature bound in the Bakry–Émery sense. Our first result using only curvature and maximal vertex degree is sharp in the case of hypercubes. The second result depends on an additional dimension bound, but is independent of the vertex degree. In particular, the second result is the first Bonnet–Myers type theorem for unbounded graph Laplacians. Moreover, our results improve diameter bounds from Fathi and Shu (Bernoulli 24(1):672–698, 2018) and Horn et al. (J für die reine und angewandte Mathematik (Crelle’s J), 2017, https://doi.org/10.1515/crelle-2017-0038) and solve a conjecture from Cushing et al. (Bakry–Émery curvature functions of graphs, 2016).

Citation

Liu, S., Münch, F., & Peyerimhoff, N. (2018). Bakry–Émery curvature and diameter bounds on graphs. Calculus of Variations and Partial Differential Equations, 57(2), Article 67. https://doi.org/10.1007/s00526-018-1334-x

• Journal Article Type: Article
• Acceptance Date: Feb 28, 2018
• Online Publication Date: Mar 15, 2018
• Publication Date: Mar 15, 2018
• Deposit Date: May 1, 2018
• Publicly Available Date: Mar 15, 2019
• Journal: Calculus of Variations and Partial Differential Equations
• Print ISSN: 0944-2669
• Electronic ISSN: 1432-0835
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 57
• Issue: 2
• Article Number: 67
• DOI: https://doi.org/10.1007/s00526-018-1334-x
• Public URL: https://durham-repository.worktribe.com/output/1359781