From error correcting codes, to atomic lattice systems, to electrical circuits, and many other topics, graph theory is a powerful tool for understanding physical systems. We explore the interplay of graph theory with a pair of familiar topics: electrical circuits and lattice models. The first major thrust of our research attempts to make broad, rigorous statements about the analysis, synthesis, and quantization of electrical circuits. The second thrust focuses on interference induced locality, the spectra of hyperbolic graphs, and other connections to spectral graph theory.