  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.
