Gluing graphs and the spectral gap: a Titchmarsh–Weyl matrix-valued function approach
Assume that two metric graphs are joined by gluing a few vertices. We investigate the behaviour of the spectral gap for the corresponding standard Laplacians. It appears that a precise answer can be given in terms of the corresponding Titchmarsh–Weyl (matrix) functions of the two subgraphs, more precisely in terms of their negative spectral subspaces. We illustrate our results by considering explicit examples.