  Kirchoff's matrix tree theorem of 1847 connects the number of spanning trees
of a graph to the spectral determinant of the discrete Laplacian [6]. 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 [11]. 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 [11].
