Quantum Hall Ground States and Regular Graphs

We show that every uniform state on the sphere is essentially a superposition of regular graphs. In addition, we develop a graph-based ansatz to construct trial FHQ ground states sharing the local properties of Jack polynomials. In particular, our graphic states have the $(k,r)$ clustering property. Moreover, a subclass of the construction is realizable as the densest zero-energy state of a model that modifies the projection Hamiltonian.