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.
28 pages; minor revision, some (Counter)examples added, to appear in: Journal of Spectral Theory