  We study the spectrum of a quantum star graph with a non-selfadjoint Robin
condition at the central vertex. We first prove that, in the high frequency
limit, the spectrum of the Robin Laplacian is close to the usual spectrum
corresponding to the Kirchhoff condition. Then, we describe more precisely the
asymptotics of the difference in terms of the Barra-Gaspard measure of the
graph. This measure depends on the arithmetic properties of the lengths of the
edges. As a by-product, this analysis provides a Weyl Law for non-selfadjoint
quantum star graphs and it gives the asymptotic behaviour of the imaginary
parts of the eigenvalues.
