Graph spectra are closely related to many applications in quantum physics: a network of quantum particles with fixed couplings can be modelled by an underlying graph, with the Hamiltonian of such system approximated by the adjacency matrix of that graph, and the energy levels and states represented by the eigenvalues and eigenvectors of the adjacency matrix. From that viewpoint, quite a few quantum physics problems can be posed in terms of the spectral properties of the graph. In this chapter we survey one particular problem which received a lot of attention recently: the existence of {\em perfect state transfer} in the network of spin-1/2 particles.