  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.
