  We study the Schr\"{o}dinger operators on a non-compact star graph with the
Coulomb-type potentials having singularities at the vertex. The convergence of
regularized Hamiltonians $H_\varepsilon$ with cut-off Coulomb potentials
coupled with $(\alpha \delta+\beta\delta')$-like ones is investigated.The 1D
Coulomb potential and the $\delta'$-potential are very sensitive to their
regularization method. The conditions of the norm resolvent convergence of
$H_\varepsilon$ depending on the regularization are established. The limit
Hamiltonians give the Schr\"{o}dinger operators with the Coulomb-type
potentials a mathematically precise meaning, ensuring the correct choice of
vertex conditions. We also describe all self-adjoint realizations of the formal
Coulomb Hamiltonians on the star graph.
