@article{KellerLenzMuenchetal.2016,
  author    = {Keller, Matthias and Lenz, Daniel and M{\"u}nch, Florentin and Schmidt, Marcel and Telcs, Andras},
  title     = {Note on short-time behavior of semigroups associated to self-adjoint operators},
  series = {Bulletin of the London Mathematical Society},
  volume    = {48},
  journal   = {Bulletin of the London Mathematical Society},
  publisher = {Oxford Univ. Press},
  address   = {Oxford},
  issn      = {0024-6093},
  doi       = {10.1112/blms/bdw054},
  pages     = {935 -- 944},
  year      = {2016},
  abstract  = {We present a simple observation showing that the heat kernel on a locally finite graph behaves for short times t roughly like t(d), where d is the combinatorial distance. This is very different from the classical Varadhan-type behavior on manifolds. Moreover, this also gives that short-time behavior and global behavior of the heat kernel are governed by two different metrics whenever the degree of the graph is not uniformly bounded.},
  language  = {en}
}
@article{KellerMugnolo2016,
  author    = {Keller, Matthias and Mugnolo, Delio},
  title     = {General Cheeger inequalities for p-Laplacians on graphs},
  series = {Theoretical ecology},
  volume    = {147},
  journal   = {Theoretical ecology},
  publisher = {Elsevier},
  address   = {Oxford},
  issn      = {0362-546X},
  doi       = {10.1016/j.na.2016.07.011},
  pages     = {80 -- 95},
  year      = {2016},
  abstract  = {We prove Cheeger inequalities for p-Laplacians on finite and infinite weighted graphs. Unlike in previous works, we do not impose boundedness of the vertex degree, nor do we restrict ourselves to the normalized Laplacian and, more generally, we do not impose any boundedness assumption on the geometry. This is achieved by a novel definition of the measure of the boundary which uses the idea of intrinsic metrics. For the non-normalized case, our bounds on the spectral gap of p-Laplacians are already significantly better for finite graphs and for infinite graphs they yield non-trivial bounds even in the case of unbounded vertex degree. We, furthermore, give upper bounds by the Cheeger constant and by the exponential volume growth of distance balls. (C) 2016 Elsevier Ltd. All rights reserved.},
  language  = {en}
}
@article{KellerMuenchPogorzelski2016,
  author    = {Keller, Matthias and M{\"u}nch, Florentin and Pogorzelski, Felix},
  title     = {Geometry and spectrum of rapidly branching graphs},
  series = {Mathematische Nachrichten},
  volume    = {289},
  journal   = {Mathematische Nachrichten},
  publisher = {Wiley-VCH},
  address   = {Weinheim},
  issn      = {0025-584X},
  doi       = {10.1002/mana.201400349},
  pages     = {1636 -- 1647},
  year      = {2016},
  abstract  = {We study graphs whose vertex degree tends to infinity and which are, therefore, called rapidly branching. We prove spectral estimates, discreteness of spectrum, first order eigenvalue and Weyl asymptotics solely in terms of the vertex degree growth. The underlying techniques are estimates on the isoperimetric constant. Furthermore, we give lower volume growth bounds and we provide a new criterion for stochastic incompleteness. (C) 2016 WILEY-VCH Verlag GmbH \& Co. KGaA, Weinheim},
  language  = {en}
}