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Materials science and the study of the electronic properties of solids are amajor field of interest in both physics and engineering. The starting point forall such calculations is single-electron, or non-interacting, band structurecalculations, and in the limit of strong on-site confinement this can bereduced to graph-like tight-binding models. In this context, bothmathematicians and physicists have developed largely independent methods forsolving these models. In this paper we will combine and present results fromboth fields. In particular, we will discuss a class of lattices which can berealized as line graphs of other lattices, both in Euclidean and hyperbolicspace. These lattices display highly unusual features including flat bands andlocalized eigenstates of compact support. We will use the methods of bothfields to show how these properties arise and systems for classifying thephenomenology of these lattices, as well as criteria for maximizing the gaps.Furthermore, we will present a particular hardware implementation usingsuperconducting coplanar waveguide resonators that can realize a wide varietyof these lattices in both non-interacting and interacting form.

Line-Graph Lattices: Euclidean and Non-Euclidean Flat Bands, and Implementations in Circuit Quantum Electrodynamics