  Perfect transfer of a quantum state through a one-dimensional chain is now
well understood, allowing one not only to decide whether a fixed Hamiltonian
achieves perfect transfer, but to design a suitable one. We are particularly
interested in being able to design, or understand the limitations imposed upon,
Hamiltonians subject to various naturally arising constraints such as a limited
coupling topology with low connectivity (specified by a graph) and type of
interaction. In this paper, we characterise the necessary and sufficient
conditions for transfer through a network, and describe some natural
consequences such as the impossibility of routing between many different
recipients for a large class of Hamiltonians, and the limitations on transfer
rate. We also consider some of the trade-offs that arise in uniformly coupled
networks (both Heisenberg and XX models) between transfer distance and the size
of the network as a consequence of the derived conditions.
