  We consider a system of qubits coupled via nearest-neighbour interaction
governed by the Heisenberg Hamiltonian. We further suppose that all coupling
constants are equal to $1$. We are interested in determining which graphs allow
for a transfer of quantum state with fidelity equal to $1$. To answer this
question, it is enough to consider the action of the Laplacian matrix of the
graph in a vector space of suitable dimension.
  Our main result is that if the underlying graph is a tree with more than two
vertices, then perfect state transfer does not happen. We also explore related
questions, such as what happens in bipartite graphs and graphs with an odd
number of spanning trees. Finally, we consider the model based on the
$XY$-Hamiltonian, whose action is equivalent to the action of the adjacency
matrix of the graph. In this case, we conjecture that perfect state transfer
does not happen in trees with more than three vertices.
