Enumeration of all superconducting circuits up to 5 nodes

Nonlinear superconducting circuits can be used as amplifiers, transducers, and qubits. Only a handful of superconducting circuits have been analyzed or built, so many high-performing configurations likely remain undiscovered. We seek to catalog this design space by enumerating all superconducting circuits - up to five nodes in size - built of capacitors, inductors, and Josephson junctions. Using graph isomorphism, we remove redundant configurations to construct a set of unique circuits. We define the concept of a "Hamiltonian class" and sort the resulting circuit Hamiltonians based on the types of variables present and the structure of their coupling. Finally, we search for novel superconducting qubits by explicitly considering all three node circuits, showing how the results of our enumeration can be used as a starting point for circuit design tasks.