  First-principles studies of the electron-phonon coupling in graphene predict
a high coupling strength for the $\sigma$ band with $\lambda$ values of up to
0.9. Near the top of the $\sigma$ band, $\lambda$ is found to be $\approx 0.7$.
This value is consistent with the recently observed kinks in the $\sigma$ band
dispersion by angle-resolved photoemission. While the photoemission intensity
from the $\sigma$ band is strongly influenced by matrix elements due to
sub-lattice interference, these effects differ significantly for data taken in
the first and neighboring Brillouin zones. This can be exploited to disentangle
the influence of matrix elements and electron-phonon coupling. A rigorous
analysis of the experimentally determined complex self-energy using
Kramers-Kronig transformations further supports the assignment of the observed
kinks to strong electron-phonon coupling and yields a coupling constant of
$0.6(1)$, in excellent agreement with the calculations.
