  We generalize the well-known "12" and "24" Theorems for reflexive polytopes
of dimension 2 and 3 to any smooth reflexive polytope. Our methods apply to a
wider category of objects, here called reflexive GKM graphs, that are
associated with certain monotone symplectic manifolds which do not necessarily
admit a toric action. As an application, we provide bounds on the Betti numbers
for certain monotone Hamiltonian spaces which depend on the minimal Chern
number of the manifold.
