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

Simplicity of eigenvalues and non-vanishing of eigenfunctions of a quantum graph