  A major application of the mathematical concept of graph in quantum mechanics
is to model networks of electrical wires or electromagnetic wave-guides. In
this paper, we address the dynamics of a particle trapped on such a network in
presence of an external electromagnetic field. We study the controllability of
the motion when the intensity of the field changes over time and plays the role
of control. From a mathematical point of view, the dynamics of the particle is
modeled by the so-called bilinear Schr\"odinger equation defined on a graph
representing the network. The main purpose of this work is to extend the
existing theory for bilinear quantum systems on bounded intervals to the
framework of graphs. To this end, we introduce a suitable mathematical setting
where to address the controllability of the equation from a theoretical point
of view. More precisely, we determine assumptions on the network and on the
potential field ensuring its global exact controllability in suitable spaces.
Finally, we discuss two applications of our results and their practical
implications to two specific problems involving a star-shaped network and a
tadpole graph.
