Can one hear the spanning trees of a quantum graph?

Kirchhoff showed that the number of spanning trees a graph is spectral determinant combinatorial Laplacian divided by vertices; we reframe this result in quantum setting. We prove Laplace operator on finite connected metric with standard (Neumann–Kirchhoff) vertex conditions determines when lengths edges are sufficiently close together. To obtain result, analyze an equilateral whose spectrum closely related to spectra discrete operators and then use continuity under perturbations edge lengths.