Three conjectures of Ostrander on digraph Laplacian eigenvectors

Abstract

Ostrander proposed three conjectures on the connections between topological properties of a weighted digraph and combinatorial properties of its Laplacian eigenvectors. We verify one of his conjectures, give counterexamples to the other two, and then seek for related valid connections and generalizations to Schrödinger operators on countable digraphs. We suggest the open question of deciding if the countability assumption can be dropped from our main results.