  We present a first-principles investigation of the phonon-induced electron
self-energy in graphene. The energy dependence of the self-energy reflects the
peculiar linear bandstructure of graphene and deviates substantially from the
usual metallic behavior. The effective band velocity of the Dirac fermions is
found to be reduced by 4-8%, depending on doping, by the interaction with
lattice vibrations. Our results are consistent with the observed linear
dependence of the electronic linewidth on the binding energy in photoemission
spectra.
