Mark Kempton

A Local Clustering Algorithm for Connection Graphs

We give a clustering algorithm for connection graphs, that is, weighted graphs in which each edge is associated with a d-dimensional rotation. The problem of interest is to identify subsets of small Cheeger ratio and which have a high level of consistency, i.e. that have small edge boundary and the rotations along any distinct paths joining two vertices are the same or within some small error factor. We use PageRank vectors as well as tools related to the Cheeger constant to give a clustering algorithm that runs in nearly linear time.

THE INVERSE EIGENVALUE AND INERTIA PROBLEMS FOR MINIMUM RANK TWO GRAPHS

Let G be an undirected graph on n vertices and let S(G) be the set of all real sym- metric n×n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. Let mr(G) denote the minimum rank of all matrices in S(G), and mr+(G) the minimum rank of all positive semidefinite matrices in S(G). All graphs G with mr(G) = 2 and mr+(G) = k are characterized; it is also noted that mr+(G) = α(G) for such graphs. This charac- terization solves the inverse inertia problem for graphs whose minimum rank is two. Furthermore, it is determined which diagonal entries are required to be zero, are required to be nonzero, or can be either for a rank minimizing matrix in S(G) when mr(G) = 2. Collectively, these results lead to a solution to the inverse eigenvalue problem for rank minimizing matrices for graphs whose minimum rank is two.

Pretty good quantum state transfer in symmetric spin networks via magnetic field

We study pretty good single-excitation quantum state transfer (i.e., state transfer that becomes arbitrarily close to perfect) between particles in symmetric spin networks, in the presence of an energy potential induced by a magnetic field. In particular, we show that if a network admits an involution that fixes at least one node or at least one link, then there exists a choice of potential on the nodes of the network for which we get pretty good state transfer between symmetric pairs of nodes. We show further that in many cases, the potential can be chosen so that it is only nonzero at the nodes between which we want pretty good state transfer. As a special case of this, we show that such a potential can be chosen on the endpoints of a spin chain to induce pretty good state transfer in chains of any length. This is in contrast to the result of Kempton et al. (Quantum Inf Comput 17(3):303–327, 2017), in which the authors show that there cannot be perfect state transfer in chains of length 4 or more, no matter what potential is chosen.