Can One Hear the Spanning Trees of a Quantum Graph?

TL;DR

This paper explores how the spectral properties of quantum graphs relate to their spanning trees, establishing that under certain conditions, the spectral determinant can determine the number of spanning trees.

Contribution

It extends Kirchhoff's classical result to quantum graphs, showing the spectral determinant's ability to determine spanning trees when edge lengths are nearly equal.

Findings

Spectral determinant determines spanning trees for nearly equilateral quantum graphs.

Spectrum of quantum graph relates closely to discrete graph spectra.

Continuity of spectral determinant under edge length perturbations is established.

Abstract

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 (Neummann-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.