  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.
