  The aim of the article is to understand the combinatorics of snake graphs by
means of linear algebra. In particular, we apply Kasteleyn's and
Temperley--Fisher's ideas about spectral properties of weighted adjacency
matrices of planar bipartite graphs to snake graphs. First we focus on snake
graphs whose set of turning vertices is monochromatic. We provide recursive
sequences to compute the characteristic polynomials; they are indexed by the
upper or the lower boundary of the graph and are determined by a neighbour
count. As an application, we compute the characteristic polynomials for
L-shaped snake graphs and staircases in terms of Fibonacci product polynomials.
Next, we introduce a method to compute the characteristic polynomials as
convergents of continued fractions. Finally, we show how to transform a snake
graph with turning vertices of two colours into a graph with the same number of
perfect matchings to which we can apply the results above.
