Title: Average Mixing Matrices of Trees and Stars
|Washington State University
|Contact Soffia Arnadottir
We define the average mixing matrix (AMM) of a continuous-time quantum walk on a graph using the orthogonal projections onto the eigenspaces of the adjacency matrix A. From there, one of the properties that has been studied is the rank of the AMM. This is easiest to do if the eigenvalues of A are simple, and we’ll review some of the results on this from Coutinho et. al. (2018). Then we’ll see how Godsil et. al. (2018) used these and other methods to prove properties about the rank of AMM for trees. A special case of tree graphs is the star graph, which does not have simple eigenvalues and thus the bounds on the rank no longer apply. We’ll show instead that the star graphs have full rank, which we proved in joint work with Dr. Judi McDonald but appears in Godsil et. al. (2018).