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Bakry–Émery curvature and diameter bounds on graphs

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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).

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Acknowledgements

We gratefully acknowledge partial support by the EPSRC Grant EP/K016687/1. FM wants to thank the German Research Foundation (DFG) for financial support.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui Province, China

    Shiping Liu

  2. Institut für Mathematik, Universität Potsdam, 14476, Potsdam, Germany

    Florentin Münch

  3. Department of Mathematical Sciences, Durham University, Durham, DH1 3LE, UK

    Norbert Peyerimhoff

Authors
  1. Shiping Liu
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  2. Florentin Münch
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  3. Norbert Peyerimhoff
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Correspondence to Florentin Münch.

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Communicated by J. Jost.

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Liu, S., Münch, F. & Peyerimhoff, N. Bakry–Émery curvature and diameter bounds on graphs. Calc. Var. 57, 67 (2018). https://doi.org/10.1007/s00526-018-1334-x

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