Kirchoff’s matrix tree theorem of 1847 connects the number of spanning trees of a graph to the spectral determinant of the discrete Laplacian [22]. Recently an analogue was obtained for quantum graphs relating the number of spanning trees to the spectral determinant of a Laplacian acting on functions on a metric graph with standard (Neumann-like) vertex conditions [20]. This result holds for quantum graphs where the edge lengths are close together. A quantum graph where the edge lengths are all equal is called equilateral. Here we consider equilateral graphs where we perturb the length of a single edge (almost equilateral graphs). We analyze the spectral determinant of almost equilateral complete graphs, complete bipartite graphs, and circulant graphs. This provides a measure of how fast the spectral determinant changes with respect to changes in an edge length. We apply these results to estimate the width of a window of edge lengths where the connection between the number of spanning trees and the spectral determinant can be observed. The results suggest the connection holds for a much wider window of edge lengths than is required in [20].
