  We compare ground states for the nonlinear Schr\"odinger equation on metric
graphs, defined as global minimizers of the action functional constrained on
the Nehari manifold, and least action solutions, namely minimizers of the
action among all solutions to the equation. In principle, four alternative
cases may take place: ground states do exist (thus coinciding with least action
solutions); ground states do not exist while least action solutions do; both
ground states and least action solutions do not exist and the levels of the two
minimizing problems coincide; both ground states and least action solutions do
not exist and the levels of the two minimizing problems are different. We show
that in the context of metric graphs all four alternatives do occur. This is
accomplished by a careful analysis of doubly constrained variational problems.
As a by-product, we obtain new multiplicity results for positive solutions on a
wide class of noncompact metric graphs.
