Quantum Walks on Oriented Graphs

Abstract

This thesis extends results about periodicity and perfect state transfer to oriented graphs. We prove that if a vertex a is periodic, then elements of the eigenvalue support lie in Z √∆ for some squarefree negative integer ∆. We find an infinite family of orientations of the complete graph that are periodic. We find an example of a graph with both perfect state transfer and periodicity that is not periodic at an integer multiple of the period, and we prove and use Gelfond-Schneider Theorem to show that every oriented graph with perfect state transfer between two vertices will have both vertices periodic. We find a complete characterization of when perfect state transfer can occur in oriented graphs, and find a new example of a graph where one vertex has perfect state transfer to multiple other vertices.