  We address the study of controllability of a closed quantum system whose
dynamical Lie algebra is generated by adjacency matrices of graphs. We
characterize a large family of graphs that renders a system controllable. The
key property is a novel graph-theoretic feature consisting of a particularly
disordered cycle structure. Disregarding efficiency of control functions, but
choosing subfamilies of sparse graphs, the results translate into
continuous-time quantum walks for universal computation.
