Volume Growth, Spectrum and Stochastic Completeness of Infinite Graphs
  We study the connections between volume growth, spectral properties and stochastic completeness of locally finite 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 graphs. 
Matthias Keller, Daniel Lenz and Radoslaw K. Wojciechowski
math.SP