SPECTRAL PROPERTIES AND SYMMETRIES OF DOUBLE COVERINGS OF GRAPHS WITH APPLICATIONS TO SUPERCONDUCTIVITY

Let M be a planar embedded graph, and let [Formula: see text] be its double covering. We count the multiplicity of the ground states of the Laplace operator on [Formula: see text] under certain symmetry constraints. The examples of interest for us are ladder-like graphs made out of n, identical rectangles. We find that in the case of an odd n, the multiplicity of the ground state is 2, and if n, is even, the ground state is simple. This result gives an answer to a conjecture by Parks on the type of phase transitions that can occur in a superconducting ladder: Parks conjectured that in the case when the magnetic field is one half fluxoid per rectangle, the phase transition would be continuous in the case of a ladder made out of two rectangles. Our result indeed implies Parks conjecture and generalizes it to any even ladder. The mathematics of this paper is a mixture of topology, symmetry arguments and comparison theorem between the eigenvalues of Laplace operators on graphs with well chosen boundary conditions.