  We analyze the degree distribution's cut-off in finite size scale-free
networks. We show that the cut-off behavior with the number of vertices $N$ is
ruled by the topological constraints induced by the connectivity structure of
the network. Even in the simple case of uncorrelated networks, we obtain an
expression of the structural cut-off that is smaller that the natural cut-off
obtained by means of extremal theory arguments. The obtained results are
explicitly applied in the case of the configuration model to recover the size
scaling of tadpoles and multiple edges.
