State transfer on graphs

If X is a graph with adjacency matrix A , then we define H ( t ) to be the operator exp ( i t A ) . We say that we have perfect state transfer in X from the vertex u to the vertex v at time τ if the u v -entry of | H ( τ ) u , v | = 1 . State transfer has been applied to key distribution in commercial cryptosystems, and it seems likely that other applications will be found. We offer a survey of some of the work on perfect state transfer and related questions. The emphasis is almost entirely on the mathematics.