In this talk we present some results about the existence of normalized ground states on noncompact metric graphs for nonlinear Schrödinger equations involving possibly both a standard power nonlinearity and delta nonlinearities located at the vertices of the graph. In the first part, we review some results when the sole standard nonlinearity is present. In the second part, we present more recent results, both when only delta nonlinearities at the vertices are considered and when standard and delta nonlinearities cohexist. In the first case, we show that the ground state problem strongly depends on the degree of periodicity of the graph, the total number of delta nonlinearities and their dislocation in the graph. In the second case, we highlight that the existence of ground states is strongly affected by the value of the mass, the relation between the powers of the two nonlinear terms and by topological and metric properties of the graph itself. These results have been obtained in collaboration with R. Adami, S. Dovetta and E. Serra (Politecnico di Torino).
