  Using density functional theory calculations we investigate the electronic
structure of graphene doped by deposition of foreign atoms. We demonstrate
that, as the charge transfer to the graphene layer increases, the band
structure of the pristine graphene sheet is substantially affected. This is
particularly relevant when Ca atoms are deposed on graphene at CaC$_{6}$
stoichiometry. Similarly to what happens in superconducting graphite
intercalated compounds, a Ca bands occurs at the Fermi level. Its hybridization
with the C states generates a strong non-linearity in one of the $\pi^{*}$
bands below the Fermi level, at energies comparable to the graphene E$_{2g}$
phonon frequency. This strong non-linearity, and not manybody effects as
previously proposed, explains the large and anisotropic values of the apparent
electron-phonon coupling measured in angular resolved photoemission.
