  We consider stationary waves on nonlinear quantum star graphs, i.e. solutions
to the stationary (cubic) nonlinear Schr\"odinger equation on a metric star
graph with Kirchhoff matching conditions at the centre. We prove the existence
of solutions that vanish at the centre of the star and classify them according
to the nodal structure on each edge (i.e. the number of nodal domains or nodal
points that the solution has on each edge). We discuss the relevance of these
solutions in more applied settings as starting points for numerical
calculations of spectral curves and put our results into the wider context of
nodal counting such as the classic Sturm oscillation theorem.
