## Abstract

Laplacian operators on finite compact metric graphs are considered under the assumption that matching conditions at graph vertices are of δ and δ' types. An infinite set of trace formulae is obtained which link together two different quantum graphs under the assumption that their spectra coincide. The general case of graph Schrödinger operators is also considered, yielding the first trace formula. Tightness of results obtained under no additional restrictions on edge lengths is demonstrated by an example. Further examples are scrutinized when edge lengths are assumed to be rationally independent. In all but one of these impossibility of isospectral configurations is ascertained.

Original language | English |
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Pages (from-to) | 43-66 |

Number of pages | 24 |

Journal | Journal of Spectral Theory |

Volume | 6 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2016 |

### Bibliographical note

Funding Information:The third author's work was partially supported by the RFBR, grant no. 12-01-00215-a and by the grant of the Ministry of Science of Ukraine.

Publisher Copyright:

© 2016 European Mathematical Society.

## Keywords

- Boundary triples
- Inverse spectral problem
- Isospectral graphs
- Laplace operator
- Quantum graphs
- Schrödinger operator
- Trace formulae

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics
- Geometry and Topology