  We study the strength of the electron-phonon interaction on Fe single adatoms
on MgO/Ag(100) based on many-body \textit{ab-initio} spin collinear
calculations. In particular, we analyze the relative importance of the
substrate and, among other results, we conclude that the interface electron
state of Ag(100) plays a prominent role in determining the electron-phonon
coupling of localized Fe electron states. The analysis of the hybridization of
the adatom with the substrate reveals qualitative differences for even or odd
coverages of MgO, affecting significantly the spectral structure and strength
of the electron-phonon coupling. Our calculations indicate that the
electron-phonon interaction is very strong for $\le$~1 layers of MgO, while it
is sharply suppressed for larger coverages, a trend that is consistent with
recent experimental findings.
