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The aim of the article is to understand the combinatorics of snake graphs by means of linear algebra. In particular, we apply Kasteleyns and Temperley--Fishers 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.
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We extend an observation due to Stong that the distribution of the number of degree $d$ irreducible factors of the characteristic polynomial of a random $n times n$ matrix over a finite field $mathbb{F}_{q}$ converges to the distribution of the number of length $d$ cycles of a random permutation in $S_{n}$, as $q rightarrow infty$, by having any finitely many choices of $d$, say $d_{1}, dots, d_{r}$. This generalized convergence will be used for the following two applications: the distribution of the cokernel of an $n times n$ Haar-random $mathbb{Z}_{p}$-matrix when $p rightarrow infty$ and a matrix version of Landaus theorem that estimates the number of irreducible factors of a random characteristic polynomial for large $n$ when $q rightarrow infty$.
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The scissors congruence conjecture for the unimodular group is an analogue of Hilberts third problem, for the equidecomposability of polytopes. Liu and Osserman studied the Ehrhart quasi-polynomials of polytopes naturally associated to graphs whose vertices have degree one or three. In this paper, we prove the scissors congruence conjecture, posed by Haase and McAllister, for this class of polytopes. The key ingredient in the proofs is the nearest neighbor interchange on graphs and a naturally arising piecewise unimodular transformation.
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