  We prove that after an arbitrarily small adjustment of edge lengths, the
spectrum of a compact quantum graph with $\delta$-type vertex conditions can be
simple. We also show that the eigenfunctions, with the exception of those
living entirely on a looping edge, can be made to be non-vanishing on all
vertices of the graph. As an application of the above result, we establish that
the secular manifold (also called "determinant manifold") of a large family of
graphs has exactly two smooth connected components.
