We investigate the spectral theory of periodic graphs which are not locally finite but carry non-negative, symmetric and summable edge weights. These periodic graphs are shown to have rather intriguing behaviour. We shall discover “partly flat bands” which are only flat for certain quasimomenta. We construct a periodic graph whose Laplacian has purely singular continuous spectrum. We prove that motion remains ballistic along at least one layer under quite general assumptions. We construct a graph whose Laplacian has purely absolutely continuous spectrum, exhibits ballistic transport, yet fails to satisfy a dispersive estimate. We believe that this class of graphs can serve as a playground to better understand exotic spectra and dynamics in the future. Based on a joint work with Joachim Kerner, Olaf Post and Mostafa Sabri (Comm. Math. Phys., 2025).
