Spectral monotonicity for Schr\"odinger operators on metric graphs

We study the influence of certain geometric perturbations on the spectra of self-adjoint Schr"odinger operators on compact metric graphs. Results are obtained for permutation invariant vertex conditions, which, amongst others, include $\delta$ and ${\delta }^{\mathrm{\prime }}$-type conditions. We show that adding edges to the graph or joining vertices changes the eigenvalues monotonically. However, the monotonicity properties may differ from what is known for the previously studied cases of Kirchhoff (or standard) and $\delta$-conditions and may depend on the signs of the coefficients in the vertex conditions.