Unbounded Laplacians on Graphs: Basic Spectral Properties and the Heat   Equation
  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. 
Matthias Keller, Daniel Lenz
math.FA math-ph math.MP math.SP