Counting invertible Schr\"odinger Operators over Finite Fields for Trees, Cycles and Complete Graphs

We count invertible Schr"odinger operators (perturbations by diagonal matrices of the adjacency matrix) over finite fieldsfor trees, cycles and complete graphs.This is achieved for trees through the definition and use of local invariants (algebraic constructions of perhapsindependent interest).Cycles and complete graphs are treated by ad hoc methods.