Mathematical Modelling of Natural Phenomena

Research Article

Unbounded Laplacians on Graphs: Basic Spectral Properties and the Heat Equation

M. Keller and D. Lenz c1

Mathematisches Institut, Friedrich Schiller Universität Jena D-07743 Jena, Germany


We discuss Laplacians on graphs in a framework of regular Dirichlet forms. We focus on phenomena related to unboundedness of the Laplacians. This includes (failure of) essential selfadjointness, absence of essential spectrum and stochastic incompleteness.

(Online publication May 12 2010)

Key Words:

  • Dirichlet forms;
  • graphs, essential self adjointness;
  • essential spectrum;
  • isoperimetric inequalities;
  • stochastic completeness

Mathematics Subject Classification:

  • 47B39;
  • 60J27


c1 Corresponding author. E-mail: