Green Operators of Networks with a new vertex

Any elliptic operator defines an automorphism on the orthogonal subspace to the eigenfunctions associated with the lowest eigenvalue, whose inverse is the orthogonal Green operator. In this study, we show that elliptic Schr\"{o}dinger operators on networks that have been obtained by adding a new vertex to a given network, can be seen as perturbations of the Schr\"{o}dinger operators on the initial network. Therefore, the Green function on the new network can be computed in terms of the Green function of the original network.