Non-Kirchhoff Vertices and Nonlinear Schrödinger Ground States on Graphs

We review some recent results and announce some new ones on the problem of the existence of ground states for the Nonlinear Schrodinger Equation on graphs endowed with vertices where the matching condition, instead of being free (or Kirchhoff’s), is non-trivially interacting. This category includes Dirac’s delta conditions, delta prime, Fulop-Tsutsui, and others.