Characterization of Green's function of discrete Schrödinger operator on a finite graph by its spanning subgraphs

The Green's function of the discrete Schödinger operator on a finite graph is considered. This setting reproduces Laplacian and signless Laplacian by adjusting appropriate potentials. We show two ways of the expression for the Green's function by using graph structures. The first way is based on the factor of the graph by subtrees which have uni-self-loops; the second way is based on that by odd unicycle graphs.