Mathematische Zeitschrift

, Volume 274, Issue 3, pp 905–932

Volume growth, spectrum and stochastic completeness of infinite graphs


  • Matthias Keller
    • Mathematisches InstitutFriedrich Schiller Universität Jena
  • Daniel Lenz
    • Mathematisches InstitutFriedrich Schiller Universität Jena
    • York College of the City University of New York

DOI: 10.1007/s00209-012-1101-1

Cite this article as:
Keller, M., Lenz, D. & Wojciechowski, R.K. Math. Z. (2013) 274: 905. doi:10.1007/s00209-012-1101-1


We study the connections between volume growth, spectral properties and stochastic completeness of locally finite weighted graphs. For a class of graphs with a very weak spherical symmetry we give a condition which implies both stochastic incompleteness and discreteness of the spectrum. We then use these graphs to give some comparison results for both stochastic completeness and estimates on the bottom of the spectrum for general locally finite weighted graphs.

Mathematics Subject Classification (2000)

Primary 39A12Secondary 58J35

Copyright information

© Springer-Verlag Berlin Heidelberg 2012