State Transfer & Strong Cospectrality in Cayley Graphs

Abstract

This thesis is a study of two graph properties that arise from quantum walks: strong cospectrality of vertices and perfect state transfer. We prove various results about these properties in Cayley graphs. We consider how big a set of pairwise strongly cospectral vertices can be in a graph. We prove an upper bound on the size of such a set in normal Cayley graphs in terms of the multiplicities of the eigenvalues of the graph. We then use this to prove an explicit bound in cubelike graphs and more generally, Cayley graphs of $Z_2^{d_1} \times Z_4^{d_2}$. We further provide an infinite family of examples of cubelike graphs (Cayley graphs of $Z_2^d$ ) in which this set has size at least four, covering all possible values of $d$. We then look at perfect state transfer in Cayley graphs of abelian groups having a cyclic Sylow-2-subgroup. Given such a group, G, we provide a complete characterization of connection sets C such that the corresponding Cayley graph for G admits perfect state transfer. This is a generalization of a theorem of Ba\v{s}i\'{c} from 2013, where he proved a similar characterization for Cayley graphs of cyclic groups