  In the present article magnetic Laplacians on a graph are analyzed. We
provide a complete description of the set of all operators which can be
obtained from a given self-adjoint Laplacian by perturbing it by magnetic
fields. In particular, it is shown that generically this set is isomorphic to a
torus. We also describe the conditions under which the operator is
unambiguously (up to unitary equivalence) defined by prescribing the magnetic
fluxes through all loops of the graph.
