  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.
