Algebraic Graph Theory Seminar - Xiaohong Zhang

Title: Oriented Cayley Graphs with all eigenvalues being integer multiples of $\sqrt{\Delta}$

Abstract:

Let $G$ be a finite abelian group. An oriented Cayley graph on $G$ is a Cayley digraph $X(G,C)$ such that $C \cap (-C)=\emptyset$. Consider the $(0,1,-1)$ skew-symmetric adjacency matrix of an oriented Cayley graph $X=X(G,C)$. We give a characterization of when all the eigenvalues of $X$ are integer multiples of $\sqrt{\Delta}$ for some square-free integer $\Delta<0$. This also characterizes oriented Cayley graphs on which the continuous quantum walks are periodic, a necessary condition for the walk to admit uniform mixing and perfect state transfer.