  We address local inelastic scattering from vibrational impurity adsorbed onto
graphene and the evolution of the local density of electron states near the
impurity from weak to strong coupling regime. For weak coupling the local
electronic structure is distorted by inelastic scattering developing peaks/dips
and steps. These features should be detectable in the inelastic electron
tunneling spectroscopy, $d^2I/dV^2$, using local probing techniques. Inelastic
Friedel oscillations distort the spectral density at energies close to the
inelastic mode. In the strong coupling limit, a local negative $U$-center forms
in the atoms surrounding the impurity site. For those atoms, the Dirac cone
structure is fully destroyed, that is, the linear energy dispersion as well as
the V-shaped local density of electron states is completely destroyed. We
further consider the effects of the negative $U$ formation and its evolution
from weak to strong coupling. The negative $U$-site effectively acts as local
impurity such that sharp resonances appear in the local electronic structure.
The main resonances are caused by elastic scattering off the impurity site, and
the features are dressed by the presence of vibrationally activated side
resonances. Going from weak to strong coupling, changes the local electronic
structure from being Dirac cone like including midgap states, to a fully
destroyed Dirac cone with only the impurity resonances remaining.
