Interlacing inequalities of Laplacian spectra on quantum star graphs

We investigate the laplacian spectrum of a quantum star graph with Robin condition applied to the interior vertices, and Dirichlet condition on the pendant vertices. Comparing with our previous results on laplacian spectrum for similar quantum graph with the Neumann-Kirchoff condition on interior vertices, we prove interlacing inequalities between the two spectra, that is the spectra from both the laplacian appear alternatingly.