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In this Thesis, we study the behavior of spectral curves of quantum graphsunder certain families of vertex conditions, called the $\delta_s$ family,which we define in this work. We focus on studying two main quantities relatedto 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 graphitself. 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 totalincrease in the eigenvalues with respect to the perturbation parameter. We usethis quantity to show that the growth of the spectral curves is uniformlybounded, and that on average it is linear, with proportionality factordetermined by the geometry of the graph. For the general $\delta_s$ family ofvertex conditions, we study a quantity known as the spectral flow, which countsthe number of oriented intersections of the spectral curves with some givenhorizontal cross section. We use this quantity to prove an index theorem whichconnects between a generalized nodal deficiency of the eigenfunctions and thestability index of a generalized Dirichlet-to-Neumann map. We also show thatthe 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.

Spectral curves of quantum graphs with $\delta_s$ type vertex conditions