We consider the Laplacians on periodic discrete graphs. The following results are obtained: 1) The Floquet-Bloch decomposition is constructed and basic properties are derived. The spectrum of the Laplacian consists of abs. continuous part plus finite number of flat bands (eigenvalues with infinite multiplicity). 2) Estimates of the Lebesgue measure of the spectrum in terms of geometric parameters of the graph are obtained. 3) It is proved that for each N>1 there exists a periodic graph on which the Laplacian has at least N flat bands. 4) We construct the perturbation theory, when we describe the spectrum of the Laplacian on the perturbed graph obtained from some graph by adding edges (periodically).
Seminar on Analysis at University of Tsukuba