Can one hear the spanning trees of a quantum graph?

Kirchhoff showed that the number of spanning trees of a graph is the spectral determinant of the combinatorial Laplacian divided by the number of vertices; we reframe this result in the quantum graph setting. We prove that the spectral determinant of the Laplace operator on a finite connected metric...

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Bibliographic Details
Published in:Letters in mathematical physics 2023-03, Vol.113 (2), Article 31
Main Authors: Harrison, Jonathan, Weyand, Tracy
Format: Article
Language:English
Subjects:
Complex Systems
Geometry
Group Theory and Generalizations
Mathematical and Computational Physics
Physics
Physics and Astronomy
Theoretical
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Summary:Kirchhoff showed that the number of spanning trees of a graph is the spectral determinant of the combinatorial Laplacian divided by the number of vertices; we reframe this result in the quantum graph setting. We prove that the spectral determinant of the Laplace operator on a finite connected metric graph with standard (Neumann–Kirchhoff) vertex conditions determines the number of spanning trees when the lengths of the edges of the metric graph are sufficiently close together. To obtain this result, we analyze an equilateral quantum graph whose spectrum is closely related to spectra of discrete graph operators and then use the continuity of the spectral determinant under perturbations of the edge lengths.
ISSN:1573-0530
1573-0530
DOI:10.1007/s11005-023-01656-0