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

TL;DR

This paper counts invertible Schr"odinger operators over finite fields for specific graph classes, using algebraic invariants for trees and specialized methods for cycles and complete graphs.

Contribution

It introduces a novel approach using local invariants to count invertible Schr"odinger operators on trees and extends methods to cycles and complete graphs.

Findings

Count of invertible Schr"odinger operators for trees, cycles, and complete graphs.

Development of local invariants for trees.

Ad hoc methods for cycles and complete graphs.

Abstract

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.