Random magnetic fields on line graphs

We study the spectral and transport properties of Schrödinger operators on line graphs with random magnetic fields. We show that it has a pure point spectrum with exponentially decaying eigenfunctions on spectral edges, whereas there appears an eigenvalue with infinite multiplicity due to the structure of line graphs. We compute the electrical conductivity which is zero on spectral edges, but is nonzero and finite on the isolated eigenvalue mentioned above. Some related problems are also discussed.