  We study spectral properties of perturbed discrete Laplacians on
two-dimensional Archimedean tilings. The perturbation manifests itself in the
introduction of non-trivial edge weights. We focus on the two lattices on which
the unperturbed Laplacian exhibits flat bands, namely the $(3.6)^2$ Kagome
lattice and the $(3.12)^2$ ``Super-Kagome'' lattice. We characterize all
possible choices for edge weights which lead to flat bands. Furthermore, we
discuss spectral consequences such as the emergence of new band gaps. Among our
main findings is that flat bands are robust under physically reasonable
assumptions on the perturbation and we completely describe the
perturbation-spectrum phase diagram. The two flat bands in the Super-Kagome
lattice are shown to even exhibit an ``all-or-nothing'' phenomenon in the sense
that there is no perturbation which can destroy only one flat band while
preserving the other.
