  It is known that up to certain pathologies, a compact metric graph with
standard vertex conditions has a Baire-generic set of choices of edge lengths
such that all Laplacian eigenvalues are simple and have eigenfunctions that do
not vanish at the vertices. We provide a new notion of strong genericity, using
subanalytic sets, that implies both Baire genericity and full Lebesgue measure.
We show that the previous genericity results for metric graphs are strongly
generic. In addition, we show that generically the derivative of an
eigenfunction does not vanish at the vertices either. In fact, we show that
generically an eigenfunction fails to satisfy any additional vertex condition.
Finally, we show that any two different metric graphs with the same edge
lengths do not share any non-zero eigenvalue, for a generic choice of lengths,
except for a few explicit cases where the graphs have a common edge-reflection
symmetry. The paper concludes by addressing three open conjectures for metric
graphs that can benefit from the tools introduced in this paper.
