Algebraic Graph Theory Seminar - Whitney Drazen

Title: K-fractional revival and approximate K-fractional revival on path graphs

Abstract:

A continuous-time quantum walk is a process on a network of quantum particles that is governed by the transition matrix U(t) = e^{-itA}, where is A is the adjacency matrix of the graph. The two-vertex phenomenon fractional revival occurs between vertices u and v at time t if the columns of U(t) corresponding to u and v are only supported on the rows indexed by those same two vertices. The well-studied perfect state transfer is a special case of this.

We will discuss K-fractional revival, the generalization of fractional revival to larger subsets of the vertices in the graph. We will also explore a related concept approximate K-fractional revival, which denotes the existence of a sequence of times for which U(t) comes arbitrarily close to achieving K-fractional revival. Both of these have spectral characterizations analogous to their two vertex cousins. With this in hand, we will exhibit families of examples of approximate K-fractional revival on path graphs.