  We consider the nonlinear Schr\"odinger equation with pure power nonlinearity
on a general compact metric graph, and in particular its stationary solutions
with fixed mass. Since the graph is compact, for every value of the mass there
is a constant solution. Our scope is to analyze (in dependence of the mass) the
variational properties of this solution, as a critical point of the energy
functional: local and global minimality, and (orbital) stability. We consider
both the subcritical regime and the critical one, in which the features of the
graph become relevant. We describe how the above properties change according to
the topology and the metric properties of the graph.
