Spectral arbitrariness for trees fails spectacularly

TL;DR

This paper demonstrates that for certain trees, the eigenvalues of all symmetric matrices with that tree as a graph are highly constrained, often with nonlinear restrictions and minimal degrees of freedom, contradicting previous expectations.

Contribution

It introduces an infinite family of trees with multiplicity lists that impose strong nonlinear eigenvalue constraints and lead to unique eigenvalue sets, expanding understanding of spectral properties of trees.

Findings

Certain trees have eigenvalues with at most 5 degrees of freedom.

Some multiplicity lists produce unique eigenvalue sets up to shifting and scaling.

The results show nonlinear eigenvalue constraints for trees, contrary to prior assumptions.

Abstract

If $G$ is a graph and $\mathbf{m}$ is an ordered multiplicity list which is realizable by at least one symmetric matrix with graph $G$, what can we say about the eigenvalues of all such realizing matrices for $\mathbf{m}$? It has sometimes been tempting to expect, especially in the case that $G$ is a tree, that any spacing of the multiple eigenvalues should be realizable. In 2004, however, F. Barioli and S. Fallat produced the first counterexample: a tree on 16 vertices and an ordered multiplicity list for which every realizing set of eigenvalues obeys a nontrivial linear constraint. We extend this by giving an infinite family of trees and ordered multiplicity lists whose sets of realizing eigenvalues are very highly constrained, with at most 5 degrees of freedom, regardless of the size of the tree in this family. In particular, we give the first examples of multiplicity lists for a tree which impose nontrivial nonlinear eigenvalue constraints and produce an ordered multiplicity list which is achieved by a unique set of eigenvalues, up to shifting and scaling.