  We show that the electron-phonon coupling in graphene, in contrast with the
non-relativistic two-dimensional electron gas, leads to shifts in the phonon
frequencies that are non-trivial functions of the electronic density. These
shifts can be measured directly in Raman spectroscopy. We show that depending
whether the chemical potential is smaller (larger) than half of the phonon
frequency, the frequency shift can negative (positive) relative to the neutral
case (when the chemical potential is at the Dirac point), respectively. We show
that the use of the static response function to calculate these shifts is
incorrect and leads always to phonon softening. In samples with many layers, we
find a shift proportional to the carrier concentration, and a splitting of the
phonon frequencies if the charge is not homogeneously distributed. We also
discuss the effects of edges in the problem.
