  We show that the spectrum of the Schrodinger operator on a finite, metric
graph determines uniquely the connectivity matrix and the bond lengths,
provided that the lengths are non-commensurate and the connectivity is simple
(no parallel bonds between vertices and no loops connecting a vertex to
itself). That is, one can hear the shape of the graph! We also consider a
related inversion problem: A compact graph can be converted into a scattering
system by attaching to its vertices leads to infinity. We show that the
scattering phase determines uniquely the compact part of the graph, under
similar conditions as above.
