We prove interlacing inequalities between spectral minimal energies of metric graphs built on Dirichlet and standard Laplacian eigenvalues, as recently introduced in Kennedy et al. (Calc Var PDE 60:61, 2021). These inequalities, which involve the first Betti number and the number of degree one vertices of the graph, recall both interlacing and other inequalities for the Laplacian eigenvalues of the whole graph, as well as estimates on the difference between the number of nodal and Neumann domains of the whole graph eigenfunctions. To this end we study carefully the principle of cutting a graph, in particular quantifying the size of a cut as a perturbation of the original graph via the notion of its rank. As a corollary we obtain an inequality between these energies and the actual Dirichlet and standard Laplacian eigenvalues, valid for all compact graphs, which complements a version for tree graphs of Friedlander’s inequalities between Dirichlet and Neumann eigenvalues of a domain. In some cases this results in better Laplacian eigenvalue estimates than those obtained previously via more direct methods.

我们证明了建立在狄利克雷和标准拉普拉斯特征值上的度量图的谱最小能量之间的交错不等式，正如肯尼迪等人最近引入的那样。（计算值功PDE 60：61，2021）。这些不等式涉及图的第一个 Betti 数和一级顶点的数量，回顾整个图的拉普拉斯特征值的交错和其他不等式，以及对节点和诺依曼域数量之间差异的估计整个图的特征函数。为此，我们仔细研究了切割图的原理，特别是通过其秩的概念将切割的大小量化为原始图的扰动. 作为推论，我们获得了这些能量与实际狄利克雷和标准拉普拉斯特征值之间的不等式，对所有紧凑图有效，它补充了弗里德兰德不等式的树图版本在域的狄利克雷和诺依曼特征值之间。在某些情况下，这会导致比以前通过更直接的方法获得的更好的拉普拉斯特征值估计。