  We define the Bloch spectrum of a quantum graph to be the collection of the
spectra of a family of Schrodinger operators parametrized by the cohomology of
the quantum graph. We show that the Bloch spectrum determines the Albanese
torus, the block structure and the planarity of the graph. It determines a
geometric dual of a planar graph. This enables us to show that the Bloch
spectrum completely determines planar 3-connected quantum graphs.
