The field of circuit QED has emerged as a rich platform for both quantum com-
putation and quantum simulation. Lattices of coplanar waveguide (CPW) resonators
realize artificial photonic materials in the tight-binding limit. Combined with strong
qubit-photon interactions, these systems can be used to study dynamical phase tran-
sitions, many-body phenomena, and spin models in driven-dissipative systems. I will
show that waveguide resonators permit the creation of unique devices which host pho-
tons in curved spaces, gapped at bands, and novel forms of qubit-qubit interaction.
I will show that graph theory is the natural language for describing these microwave
photonic systems and that combining this mathematical tool box with the physics of
lattice systems sheds light both on the kinds of devices that are achievable in circuit
QED and present preliminary data on the development of a new generation of CPW
lattice devices with unconventional band structures.