  We consider a nonlinear Schr\"odinger equation with focusing nonlinearity of
power type on a star graph ${\mathcal G}$, written as $ i \partial_t \Psi (t) =
H \Psi (t) - |\Psi (t)|^{2\mu}\Psi (t)$, where $H$ is the selfadjoint operator
which defines the linear dynamics on the graph with an attractive $\delta$
interaction, with strength $\alpha < 0$, at the vertex. The mass and energy
functionals are conserved by the flow. We show that for $0<\mu<2$ the energy at
fixed mass is bounded from below and that for every mass $m$ below a critical
mass $m^*$ it attains its minimum value at a certain $\hat \Psi_m \in H^1(\GG)
$, while for $m>m^*$ there is no minimum. Moreover, the set of minimizers has
the structure ${\mathcal M}={e^{i\theta}\hat \Psi_m, \theta\in \erre}$.
Correspondingly, for every $m<m^*$ there exists a unique $\omega=\omega(m)$
such that the standing wave $\hat\Psi_{\omega}e^{i\omega t} $ is orbitally
stable. To prove the above results we adapt the concentration-compactness
method to the case of a star graph. This is non trivial due to the lack of
translational symmetry of the set supporting the dynamics, i.e. the graph. This
affects in an essential way the proof and the statement of
concentration-compactness lemma and its application to minimization of
constrained energy. The existence of a mass threshold comes from the
instability of the system in the free (or Kirchhoff's) case, that in our
setting corresponds to $\al=0$.
