Graph Spectra and Quantum Walks

If A is the adjacency matrix of a graph X, then the unitary operators defined by U(t) = exp(-itA) define what physicists call a continuous quantum walk. A basic problem is to relate the physical properties of this system to features of the underlying graph. One important question is whether for a given graph there are distinct vertices a and b and a time t such that |U(t)_{a,b}|=1 (If this happens we have perfect state transfer).

My talk will provide an introduction to perfect state transfer, with an emphasis on a number of connections with classical (or, at least, old) problems in graph theory.