Quasi-isospectrality on Quantum Graphs

@article{Rueckriemen2012QuasiisospectralityOQ,
  title={Quasi-isospectrality on Quantum Graphs},
  author={Ralf Rueckriemen},
  journal={The Journal of Geometric Analysis},
  year={2012},
  volume={25},
  pages={306-316}
}
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 completely identical. 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. 
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