An uncertainty principle and lower bounds for the Dirichlet Laplacian on   graphs
  We prove a quantitative uncertainty principle at low energies for the Laplacian on fairly general weighted graphs with a uniform explicit control of the constants in terms of geometric quantities. A major step consists in establishing lower bounds for Dirichlet eigenvalues in terms of the geometry. 
Daniel Lenz, Peter Stollmann, Gunter Stolz
math.FA