On Spectra of Metric Tree Graphs
Tyler Chamberlain
TL;DR
This paper provides a comprehensive description of the singular locus of the secular manifold for metric tree graphs, confirming a conjecture and exploring applications related to their spectral properties.
Contribution
It offers a complete characterization of the singular locus for metric tree graphs, validating a conjecture by Colin de Verdière and discussing related applications.
Findings
Confirmed the conjecture of Colin de Verdière regarding the singular locus.
Provided a complete description of the secular manifold's singularities for tree graphs.
Discussed applications in studying the spectral behavior of metric tree graphs.
Abstract
The secular manifold and its singularities are intimately related to the spectra of metric graphs . In this paper, we present a complete description of the singular locus for tree graphs, and confirm that it agrees with a conjecture of Colin de Verdi\`ere. We also discuss numerous applications toward studying the behavior of metric tree graphs.
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Taxonomy
TopicsGraph Labeling and Dimension Problems · Graph theory and applications · Graph Theory and Algorithms