The Nodal Count {0, 1, 2, 3,...} Implies The Graph is a Tree

TL;DR

This paper proves that if all eigenfunctions of a graph have a number of zeros matching their eigenvalue index, then the graph must be a tree, extending classical oscillation theorems to discrete and metric graphs.

Contribution

It establishes the converse of Sturm's oscillation theorem for both discrete and metric graphs, linking nodal counts to the graph's structure as a tree.

Findings

Eigenfunctions with n-1 zeros imply the graph is a tree.

Develops a notion of discretized metric graphs and relates their nodal counts.

Shows that all eigenvalues cannot exhibit diamagnetic behavior.

Abstract

Sturm's oscillation theorem states that the n-th eigenfunction of a Sturm-Liouville operator on the interval has n-1 zeros (nodes). This result was generalized for all metric tree graphs and an analogous theorem was proven for discrete tree graphs. We prove the converse theorems for both discrete and metric graphs. Namely, if for all n, the n-th eigenfunction of the graph has n-1 zeros then the graph is a tree. Our proofs use a recently obtained connection between the graph's nodal count and the magnetic stability of its eigenvalues. In the course of the proof we show that it is not possible for all (or even almost all, in the metric case) the eigenvalues to exhibit a diamagnetic behaviour. In addition, we develop a notion of 'discretized' versions of a metric graph and prove that their nodal counts are related to this of the metric graph.