Spectral curves of quantum graphs with \(\delta_s\) type vertex conditions

In this Thesis, we study the behavior of spectral curves of quantum graphs under certain families of vertex conditions, called the \(\delta_s\) family, which we define in this work. We focus on studying two main quantities related to the spectral curves, known as the Robin-Neumann gap and the spectral flow. We show that these quantities hold information about the the spectral curves, the behavior of the corresponding eigenfunctions, and the geometry of the graph itself. For a specific subset of the \(\delta_s\) family which is known as the \(\delta\) family, we study the Robin-Neumann gap, which measures the total increase in the eigenvalues with respect to the perturbation parameter. We use this quantity to show that the growth of the spectral curves is uniformly bounded, and that on average it is linear, with proportionality factor determined by the geometry of the graph. For the general \(\delta_s\) family of vertex conditions, we study a quantity known as the spectral flow, which counts the number of oriented intersections of the spectral curves with some given horizontal cross section. We use this quantity to prove an index theorem which connects between a generalized nodal deficiency of the eigenfunctions and the stability index of a generalized Dirichlet-to-Neumann map. We also show that the spectral flow holds information about the graph topology. Parts of the thesis are based on joint work with Ram Band, Marina Prokhorova, Holger Schanz, and Uzy Smilansky.