  We investigate the existence of ground states of prescribed mass, for the
nonlinear Schroedinger energy on a noncompact metric graph G. While in some
cases the topology of G may rule out or, on the contrary, guarantee the
existence of ground states of any given mass, in general also some metric
properties of G, and their quantitative relation with the actual value of the
prescribed mass, are relevant as concerns the existence of ground states. This
may give rise to interesting phase transitions from nonexistence to existence
of ground states, when a certain quantity reaches a critical threshold.
