Approximations of quantum-graph vertex couplings by singularly scaled rank-one operators

We investigate approximations of the vertex coupling on a star-shaped graph by families of operators with singularly scaled rank-one interactions. We find a family of vertex couplings, generalizing the $\delta'$-interaction on the line, and show that with a suitable choice of the parameters they can be approximated in this way in the norm-resolvent sense. We also analyze spectral properties of the involved operators and demonstrate convergence of the corresponding on-shell scattering matrices.