Counting Invertible Schrödinger Operators over Finite Fields for Trees, Cycles and Complete Graphs

Abstract

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