  We revisit the Anderson localization problem on Bethe lattices, putting in
contact various aspects which have been previously only discussed separately.
For the case of connectivity 3 we compute by the cavity method the density of
states and the evolution of the mobility edge with disorder. Furthermore, we
show that below a certain critical value of the disorder the smallest
eigenvalue remains delocalized and separated by all the others (localized) ones
by a gap. We also study the evolution of the mobility edge at the center of the
band with the connectivity, and discuss the large connectivity limit.
