Nonlinear dynamics on graphs has rapidly become a topical issue with many physical applications, ranging from nonlinear optics to Bose–Einstein condensation. In the first part of the talk, we introduce the problem of the existence of ground states at fixed mass for a focusing nonlinear Schrödinger equation with a standard nonlinearity of power type, highlighting how the topology of the graph can affect the existence results. In the second part, we study the same minimization problem for a nonlinear Schrödinger equation with two focusing power–type nonlinear terms: a pointwise one, located at the vertices of the graph, and a standard one. We show that existence and non–existence results strongly depend on the interplay between the two nonlinearities and on the topological and metric properties of the graph.