We consider the forward and inverse scattering problems on perturbed periodic graphs. A physically important example is the graphen, for which there are two closely related mathematical models. The first one (the discrete model or the vertex model) deals with the propagation waves restricted on vertices of hexagonal lattices, and the second one (the quantum graph or the edge model) describes the waves governed by the 1-dimensional Schroedinger equation on the edges. We develop the spectral theory for these graph Laplacians by constructing a complete system of generalized eigenfunctions, studying their behavior at infinity and defining the S-matrix. The main aim is to solve the inverse scattering problem. Assuming that perturbations are confined to a finite part of the graph, we show that for the vertex mode
(1) the S-matrix of a fixed energy determines the potential,
(2) the S-matrix of a fixed energy determines the convex hull of defects of the lattice,
(3) the S-matrices for all energies determines the perturbation as a planar graph.
For the edge model we show that
(4) the S-matrices for all energies determine the potentials on all edges provided they are symmetric with respect to the center of the edge.
Our second problem is the convergence of the discrete system to the continuous model when the mesh size tends to 0. We develop a general framework for a class of lattices including the square, triangular, hexagonal, graphite and Kagome lattices etc. to derive Shroedinger equations or Dirac equations
