Geodesic
Singular polynomials of generalized Kasteleyn matrices
Journal of Algebraic Combinatorics, Tome 16 (2002) no. 2, pp. 195-207.

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Summary: Kasteleyn counted the number of domino tilings of a rectangle by considering a mutation of the adjacency matrix: a Kasteleyn matrix K. In this paper we present a generalization of Kasteleyn matrices and a combinatorial interpretation for the coefficients of the characteristic polynomial of $K$K* (which we call the singular polynomial), where $K$ is a generalized Kasteleyn matrix for a planar bipartite graph. We also present a $q$-version of these ideas and a few results concerning tilings of special regions such as rectangles.
Keywords: domino tilings, dimers, kasteleyn matrix, singular values 
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     author = {Saldanha, Nicolau C.},
     title = {Singular polynomials of generalized {Kasteleyn} matrices},
     journal = {Journal of Algebraic Combinatorics},
     pages = {195--207},
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     number = {2},
     year = {2002},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_2002__16_2_a1/}
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Saldanha, Nicolau C. Singular polynomials of generalized Kasteleyn matrices. Journal of Algebraic Combinatorics, Tome 16 (2002) no. 2, pp. 195-207. http://geodesic.mathdoc.fr/item/JAC_2002__16_2_a1/