Quasi-isospectrality on quantum graphs

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.