  Many-body perturbation theory at the $G_0W_0$ level is employed to study the
electronic properties of poly(\emph{para}-phenylene) (PPP) on graphene.
Analysis of the charge density and the electrostatic potential shows that the
polymer-surface interaction gives rise to the formation of only weak surface
dipoles with no charge transfer between the polymer and the surface. In the
local-density approximation (LDA) of density-functional theory, the band
structure of the combined system appears as a superposition of the eigenstates
of its constituents. Consequently, the LDA band gap of PPP remains unchanged
upon adsorption onto graphene. $G_0W_0$ calculations, however, renormalize the
electronic levels of the weakly physisorbed polymer. Thereby, its band gap is
considerably reduced compared to that of the isolated PPP chain. This effect
can be understood in terms of image charges induced in the graphene layer,
which allows us to explain the quasi-particle gap of PPP versus
polymer-graphene distance by applying a classical image-potential model. For
distances below 4.5 {\AA}, however, deviations from this simple classical model
arise which we qualitatively explain by taking into account the polarizablity
of the adsorbate. For a quantitative description with predictive power,
however, we emphasize the need for an accurate ab-initio description of the
electronic structure for weakly coupled systems at equilibrium bonding
distances.
