Fluxes, Laplacians, and Kasteleyn’s theorem. (English) Zbl 0787.05083

Lieb, Elliott H.; Loss, Michael

Duke Math. J. 71, No. 2, 337-363 (1993).

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.

Reviewer: H.N.V.Temperley (Langport)

MSC:

05C90	Applications of graph theory
82B20	Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
05C50	Graphs and linear algebra (matrices, eigenvalues, etc.)
05C10	Planar graphs; geometric and topological aspects of graph theory
81V10	Electromagnetic interaction; quantum electrodynamics
05C05	Trees
05B40	Combinatorial aspects of packing and covering
05B20	Combinatorial aspects of matrices (incidence, Hadamard, etc.)

Keywords:

Laplacians; crystal lattice; magnetic field; matrices; determinants; eigenvalues; Kasteleyn’s theorem; dimer coverings; planar graph

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