Given λ > 0 and p > 2, we present a complete classification of the positive H 1-solutions of the equation on the -metric graph (consisting of two unbounded edges and a terminal edge of length , all joined together at a single vertex). This study implies, in particular, the uniqueness of action ground states. Moreover, for , the notions of action and energy ground states do not coincide and energy ground states are not unique. In the L 2-supercritical case p > 6, we prove that, for and , action ground states are orbitally unstable for the flow generated by the associated time-dependent NLS equation . Finally, we provide numerical evidence of the uniqueness of energy ground states for and of the existence of both stable and unstable action ground states for .