  The aim of this work is to study the controllability of the bilinear
Schr\"odinger equation on compact graphs. In particular, we consider the
equation (BSE) $i\partial_t\psi=-\Delta\psi+u(t)B\psi$ in the Hilbert space
$L^2(\mathscr{G},\mathbb{C})$, with $\mathscr{G}$ being a compact 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
provide a new technique leading to the global exact controllability of the
(BSE) in $D(|\Delta|^{s/2})$ with $s\geq 3$. Afterwards, we introduce the
"energetic controllability", a weaker notion of controllability useful when the
global exact controllability fails. In conclusion, we develop some applications
of the main results involving for instance star graphs.
