Quantum graphs and dimensional crossover: the honeycomb

We summarize features and results on the problem of the existence of Ground States for the Nonlinear Schr"odinger 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.