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The aim of the article is to understand the combinatorics of snake graphs bymeans of linear algebra. In particular, we apply Kasteleyn's andTemperley--Fisher's ideas about spectral properties of weighted adjacencymatrices of planar bipartite graphs to snake graphs. First we focus on snakegraphs whose set of turning vertices is monochromatic. We provide recursivesequences to compute the characteristic polynomials; they are indexed by theupper or the lower boundary of the graph and are determined by a neighbourcount. As an application, we compute the characteristic polynomials forL-shaped snake graphs and staircases in terms of Fibonacci product polynomials.Next, we introduce a method to compute the characteristic polynomials asconvergents of continued fractions. Finally, we show how to transform a snakegraph with turning vertices of two colours into a graph with the same number ofperfect matchings to which we can apply the results above.

Snake graphs and their characteristic polynomials