  We consider the problem of uniqueness of ground states of prescribed mass for
the Nonlinear Schr\"odinger Energy with power nonlinearity on noncompact metric
graphs. We first establish that the Lagrange multiplier appearing in the NLS
equation is constant on the set of ground states of mass $\mu$, up to an at
most countable set of masses. Then we apply this result to obtain uniqueness of
ground states on two specific noncompact graphs. Finally we construct a graph
that admits at least two ground states with the same mass having different
Lagrange multipliers. Our proofs are based on careful variational arguments and
rearrangement techniques, and hold both for the subcritical range $p\in(2,6)$
and in the critical case $p = 6$.
