  In this work, we consider the bilinear Schr\"odinger equation
$i\partial_t\psi=-\Delta\psi+u(t)B\psi$ in the Hilbert space
$L^2(\mathcal{G},\mathbb{C})$ with $\mathcal{G}$ an infinite graph. The
Laplacian $-\Delta$ is equipped with self-adjoint boundary conditions, $B$ is a
bounded symmetric operator and $u\in L^2((0,T),\mathbb{R})$ with $T>0$. We
study the well-posedness in suitable subspaces of $D(|\Delta|^{3/2})$ preserved
by the dynamics despite the dispersive behaviour of the equation. In such
spaces, we study the global exact controllability and the
{\virgolette{energetic controllability}}. We provide examples involving for
instance infinite tadpole graphs.
