  In this paper, we investigate spectral properties of discrete Laplacians. Our
study is based on the Hardy inequality and the use of super-harmonic functions.
We recover and improve lower bounds for the bottom of the spectrum and of the
essential spectrum. In some situation, we obtain Weyl asymptotics for the
eigenvalues. We also provide a probabilistic representation of super-harmonic
functions. Using coupling arguments, we set comparison results for the bottom
of the spectrum, the bottom of the essential spectrum and the stochastic
completeness of different discrete Laplacians. The class of weakly spherically
symmetric graphs is also studied in full detail.
