  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 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.
