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.
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