On Spectra of Metric Tree Graphs

The secular manifold ${\mathrm{\Sigma }}_G$ and its singularities are intimately related to the spectra of metric graphs $(G,\mathrm{\ell })$. 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 Verdière. We also discuss numerous applications toward studying the behavior of metric tree graphs.