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, [16, 12]. 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.
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The author would like to thank Peter Sarnak, Mark Goresky, Karen Uhlenbeck, Pavel Kurasov, and Ram Band for insightful discussions, important remarks and relevant references. The author was supported by the Ambrose Monell Foundation and the Institute for Advanced Study.
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Alon, L. Generic Laplacian eigenfunctions on metric graphs. JAMA (2023). https://doi.org/10.1007/s11854-023-0308-x