Fluxes, Laplacians, and Kasteleyn's theorem (Q1312784)

The authors consider some problems in graph theory arising from the problem of electrons in a crystal lattice interacting with a magnetic field. This leads to the consideration of matrices whose elements are of the form \(t_{xy}=| t_{xy}| e^{i\theta(x,y)}\). Various theorems about the determinants and eigenvalues of such matrices are proved, notably Kasteleyn's theorem that enables us to enumerate the dimer coverings of any planar graph.