  We perform a phenomenological analysis of the problem of the electronic
doping of a graphene sheet by deposited transition metal atoms, which aggregate
in clusters. The sample is placed in a capacitor device such that the
electronic doping of graphene can be varied by the application of a gate
voltage and such that transport measurements can be performed via the
application of a (much smaller) voltage along the graphene sample, as reported
in the work of Pi et al. [Phys. Rev. B 80, 075406 (2009)]. The analysis allows
us to explain the thermodynamic properties of the device, such as the level of
doping of graphene and the ionisation potential of the metal clusters in terms
of the chemical interaction between graphene and the clusters. We are also
able, by modelling the metallic clusters as perfect conducting spheres, to
determine the scattering potential due to these clusters on the electronic
carriers of graphene and hence the contribution of these clusters to the
resistivity of the sample. The model presented is able to explain the
measurements performed by Pi et al. on Pt-covered graphene samples at the
lowest metallic coverages measured and we also present a theoretical argument
based on the above model that explains why significant deviations from such a
theory are observed at higher levels of coverage.
