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.

众所周知，在某些病理情况下，具有标准顶点条件的紧凑度量图具有贝尔通用的边长度选择集，使得所有拉普拉斯特征值都是简单的并且具有不会在顶点处消失的特征函数，[16, 12 ]。我们使用亚分析集提供了强通用性的新概念，这意味着贝尔通用性和完整的勒贝格测度。我们表明，先前的度量图的通用性结果具有很强的通用性。此外，我们还表明，一般来说，本征函数的导数在顶点处也不会消失。事实上，我们表明，一般来说，特征函数无法满足任何附加的顶点条件。最后，我们表明，对于长度的通用选择，具有相同边长度的任何两个不同的度量图不共享任何非零特征值，除了一些明确的情况外，这些图具有共同的边缘反射对称性。本文最后提出了度量图的三个开放猜想，这些猜想可以从本文介绍的工具中受益。