Non-existence of co-spectral simple connected graphs with small number of edges

Authors

  • O. Boyko K.D. Ushinsky South Ukrainian National Pedagogical University, 26 Staroportofrankivska str., 65026, Odesa, Ukraine
  • D. Kaliuzhnyi-Verbovetskyi K.D. Ushinsky South Ukrainian National Pedagogical University, 26 Staroportofrankivska str., 65026, Odesa, Ukraine https://orcid.org/0000-0002-7411-3740
  • V. Pivovarchik K.D. Ushinsky South Ukrainian National Pedagogical University, 26 Staroportofrankivska str., 65026, Odesa, Ukraine; University of Vaasa, Wolffintie 32, 65200, Vaasa, Finland
https://doi.org/10.15330/cmp.18.1.99-110

Keywords:

graph, edge, vertex, eigenvalue, potential, adjacency matrix, characteristic function
Published online: 2026-05-13

Abstract

We show that the spectrum of the Sturm-Liouville problem on a connected simple equilateral graph with the Dirichlet boundary conditions at the pendant vertices is related with the spectrum of the discrete Laplacian of the corresponding combinatorial graph. It enables us to compare the spectra of discrete Laplacians to find co-spectral combinatorial graphs and finally co-spectral quantum graphs. Using this method we prove that there are no co-spectral (in our sense) graphs with the number of edges less or equal 7. Thus, in this case the inverse problem of recovering the shape of a quantum graph possesses a unique solution.

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How to Cite
(1)
Boyko, O.; Kaliuzhnyi-Verbovetskyi, D.; Pivovarchik, V. Non-Existence of Co-Spectral Simple Connected Graphs With Small Number of Edges. Carpathian Math. Publ. 2026, 18, 99-110.