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arXiv · 2301.11073

Spectral arbitrariness for trees fails spectacularly

Shaun M. Fallat·H. Tracy Hall·Rupert H. Levene·Seth A. Meyer·Shahla Nasserasr·Polona Oblak·Helena Šmigoc
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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.

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Shaun M. Fallat, H. Tracy Hall, Rupert H. Levene, Seth A. Meyer, Shahla Nasserasr, Polona Oblak, Helena Šmigoc. 2023-01-26. Spectral arbitrariness for trees fails spectacularly. https://arxiv.org/abs/2301.11073

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