Laplacians on periodic discrete graphs

We consider Laplacians on $\Z^2$-periodic discrete graphs. The following results are obtained: 1) The Floquet-Bloch decomposition is constructed and basic properties are derived. 2) The estimates of the Lebesgue measure of the spectrum in terms of geometric parameters of the graph are obtained. 3) The spectrum of the Laplacian is described, when the so-called fundamental graph consists of one or two vertices and any number of edges. 4) We consider the hexagonal lattice perturbed by adding one edge to the fundamental graph. There exist two cases: a) if the perturbed hexagonal lattice is bipartite, then the spectrum of the perturbed Laplacian coincides with the spectrum $[-1,1]$ for the unperturbed case, b) if the perturbed hexagonal lattice is not bipartite, then there is a gap in the spectrum of the perturbed Laplacian. Moreover, some deeper results are obtained for the perturbation of the square lattice.