  We investigate the existence of ground states with prescribed mass for the
NLS energy with combined $L^2$-critical and subcritical nonlinearities, on a
general non-compact metric graph $\mathcal{G}$. The interplay between the
different nonlinearities creates new phenomena with respect to purely critical
or subcritical problems on graphs; from a different perspective, topological
and metric properties of the underlying graph drastically influence existence
and non-existence of ground states with respect to the analogue problem on the
real line.
