The Spectrum of Magnetic Schrödinger Operators on a Graph with Periodic Structure

For discrete magnetic Schrodinger operators on covering graphs of a finite graph, we investigate two spectral properties: (1) the relationship between the spectrum of the operator on the covering graph and that on a finite graph, (2) the analyticity of the bottom of the spectrum with respect to magnetic flow. Also we compute the second derivative of the bottom of the spectrum and represent it in terms of geometry of a graph.