Ever since the first observation of Bose-Einstein condensation in the nineties, ultracold quantum gases have been the subject of intense research, providing a unique tool to understand the behavior of matter governed by the laws of quantum mechanics. Ultracold bosonic atoms loaded in an optical lattice are usually described by the Bose-Hubbard model or a variant of it. In addition to the common insulating and superfluid phases, other phases (like density waves and supersolids) may show up in the presence of a short-range interparticle repulsion and also depending on the geometry of the lattice. We herein explore this possibility, using the graph of a convex polyhedron as lattice and playing with the coordination of nodes to promote the wanted finite-size ordering. To accomplish the job we employ the method of decoupling approximation, whose efficacy is tested in one case against exact diagonalization. We report zero-temperature results for two Catalan solids, the tetrakis hexahedron and the pentakis dodecahedron, for which a thorough ground-state analysis reveals the existence of insulating phases with polyhedral order and a widely extended supersolid region. The key to this outcome is the unbalance in coordination between inequivalent nodes of the graph. The predicted phases can be probed in systems of ultracold atoms using programmable holographic optical tweezers.
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