We consider a particle constrained in a graph structure and excited by an external controlling field. Its dynamics is modeled by the bilinear Schrödinger equation i∂t ψ = −∆ψ + u(t)Bψ in the Hilbert space L2(G , C) where G is the graph. The Laplacian −∆ is equipped with self-adjoint boundary conditions. The action of the field is represented by the bounded symmetric operator B and by the control function u ∈ L2((0,T),R) with T > 0, which accounts its intensity. The exact controllability of the bilinear Schrödinger equation on bounded intervals was widely studied in literature. Nevertheless, the bilinear Schrödinger equation on graphs is in general a more delicate matter and it was only studied on compact networks. Up to our knowledge, the controllability on infinite graphs is still an open problem. The main reason can be found on the dispersive phenomena characterizing the equation (not considering the difficulties already appearing on compact graphs). A peculiarity of the Schrödinger equation is the loss of localization of the wave packets during the evolution, the dispersion. This effect can be measured by L ∞ -time decay. In this talk, we present the bilinear Schrödinger equation on infinite graphs. In par- ticular, we show the existence of suitable subspaces of L 2 (G , C) where the equation is well-posed. In such spaces, we define assumptions on the structure of the graph and on the control field such that the global exact controllability is guaranteed. The result leads to the so-called “energetic controllability”.