  Consider two quantum graphs with the standard Laplace operator and non-Robin
type boundary conditions at all vertices. We show that if their
eigenvalue-spectra agree everywhere aside from a sufficiently sparse set, then
the eigenvalue-spectra and the length-spectra of the two quantum graphs are
identical, with the possible exception of the multiplicity of the eigenvalue
zero. Similarly if their length-spectra agree everywhere aside from a
sufficiently sparse set, then the quantum graphs have the same
eigenvalue-spectrum and length-spectrum, again with the possible exception of
the eigenvalue zero.
