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

Robustness of flat bands on the perturbed Kagome and the perturbed Super-Kagome lattice