We summarize features and results on the problem of the existence of Ground States for the Nonlinear Schrödinger Equation on doubly-periodic metric graphs. We extend the results known for the two-dimensional square grid graph to the honeycomb, made of infinitely-many identical hexagons. Specifically, we show how the coexistence between one-dimensional and two-dimensional scales in the graph structure leads to the emergence of threshold phenomena known as dimensional crossover.

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