  Angle-resolved photoemission spectroscopy reveals pronounced kinks in the
dispersion of the sigma band of graphene. Such kinks are usually caused by the
combination of a strong electron-boson interaction and the cut-off in the
Fermi-Dirac distribution. They are therefore not expected for the $\sigma$ band
of graphene that has a binding energy of more than 3.5 eV. We argue that the
observed kinks are indeed caused by the electron-phonon interaction, but the
role of the Fermi-Dirac distribution cutoff is assumed by a cut-off in the
density of $\sigma$ states. The existence of the effect suggests a very weak
coupling of holes in the $\sigma$ band not only to the $\pi$ electrons of
graphene but also to the substrate electronic states. This is confirmed by the
presence of such kinks for graphene on several different substrates that all
show a strong coupling constant of lambda=1.
