# Visibility of quantum graph spectrum from the vertices

We investigate the relation between the eigenvalues of the Laplacian with Kirchhoff vertex

conditions on a finite metric graph and a corresponding Titchmarsh–Weyl function (a

parameter-dependent Neumann-to-Dirichlet map). We give a complete description of all real

resonances, including multiplicities, in terms of the edge lengths and the connectivity of the

graph, and apply it to characterize all eigenvalues which are visible for the Titchmarsh–Weyl

function.

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