  In some previous works, the analytic structure of the spectrum of a quantum
graph operator as a function of the vertex conditions and other parameters of
the graph was established. However, a specific local coordinate chart on the
Grassmanian of all possible vertex conditions was used, thus creating an
erroneous impression that something ``wrong'' can happen at the boundaries of
the chart. Here we show that the analyticity of the corresponding ``dispersion
relation'' holds over the whole Grassmannian, as well as over other parameter
spaces.
  We also address the Dirichlet-to-Neumann (DtN) technique of relating quantum
and discrete graph operators, which allows one to transfer some results from
the discrete to the quantum graph case, but which has issues at the Dirichlet
spectrum. We conclude that this difficulty, as in the first part of the paper,
stems from the use of specific coordinates in a Grassmannian and show how to
avoid it to extend some of the consequent results to the general situation.
