Interlacing and Friedlander-type inequalities for spectral minimal partitions of metric graphs

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, arXiv:2005.01126]. These inequalities, which involve the first Betti number degree one vertices graph, recall both other for eigenvalues whole well estimates difference nodal Neumann domains graph eigenfunctions. To this end we study carefully principle cutting a particular quantifying size cut perturbation original via notion its rank. As corollary obtain an inequality these actual valid all compact graphs, complements version tree Friedlander's domain. In some cases results better eigenvalue than those obtained previously more direct methods.