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Bakry–Émery curvature and diameter bounds on graphs

  • Shiping Liu
  • Florentin MünchEmail author
  • Norbert Peyerimhoff
Article

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).

Mathematics Subject Classification

53C21 05C81 35K05 

Notes

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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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Shiping Liu
    • 1
  • Florentin Münch
    • 2
    Email author
  • Norbert Peyerimhoff
    • 3
  1. 1.School of Mathematical SciencesUniversity of Science and Technology of ChinaHefeiChina
  2. 2.Institut für MathematikUniversität PotsdamPotsdamGermany
  3. 3.Department of Mathematical SciencesDurham UniversityDurhamUK

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