We consider a Schr\"odinger particle on a graph consisting of $\,N\,$ links
joined at a single point. Each link supports a real locally integrable
potential $\,V_j\,$; the self--adjointness is ensured by the $\,\delta\,$ type
boundary condition at the vertex. If all the links are semiinfinite and ideally
coupled, the potential decays as $\,x^{-1-\epsilon}$ along each of them, is
non--repulsive in the mean and weak enough, the corresponding Schr\"odinger
operator has a single negative eigenvalue; we find its asymptotic behavior. We
also derive a bound on the number of bound states and explain how the
$\,\delta\,$ coupling constant may be interpreted in terms of a family of
squeezed potentials.