@article{LiuMuenchPeyerimhoff2018,
  author    = {Liu, Shiping and M{\"u}nch, Florentin and Peyerimhoff, Norbert},
  title     = {Bakry-Emery curvature and diameter bounds on graphs},
  series = {Calculus of variations and partial differential equations},
  volume    = {57},
  journal   = {Calculus of variations and partial differential equations},
  number    = {2},
  publisher = {Springer},
  address   = {Heidelberg},
  issn      = {0944-2669},
  doi       = {10.1007/s00526-018-1334-x},
  pages     = {9},
  year      = {2018},
  abstract  = {We prove finiteness and diameter bounds for graphs having a positive Ricci-curvature bound in the Bakry-{\´E}mery 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 f{\"u}r 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-{\´E}mery curvature functions of graphs, 2016).},
  language  = {en}
}
@article{KellerLiuPeyerimhoff2021,
  author    = {Keller, Matthias and Liu, Shiping and Peyerimhoff, Norbert},
  title     = {A note on eigenvalue bounds for non-compact manifolds},
  series = {Mathematische Nachrichten},
  volume    = {294},
  journal   = {Mathematische Nachrichten},
  number    = {6},
  publisher = {Wiley-VCH},
  address   = {Weinheim},
  issn      = {0025-584X},
  doi       = {10.1002/mana.201900209},
  pages     = {1134 -- 1139},
  year      = {2021},
  abstract  = {In this article we prove upper bounds for the Laplace eigenvalues lambda(k) below the essential spectrum for strictly negatively curved Cartan-Hadamard manifolds. Our bound is given in terms of k(2) and specific geometric data of the manifold. This applies also to the particular case of non-compact manifolds whose sectional curvature tends to -infinity, where no essential spectrum is present due to a theorem of Donnelly/Li. The result stands in clear contrast to Laplacians on graphs where such a bound fails to be true in general.},
  language  = {en}
}