Surgery of graphs and spectral gap: Titchmarsh-Weyl operator-function approach

Assume that two metric graphs are joined together by gluing fewv ertices. 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 findings by considering explicit examples
VR 621-2013-4973