  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.
