### Xiaohong Zhang

Affiliation: University of Waterloo, Ontario

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.

You can watch the talk here, and the slides are available here.

This talk was given on March 28th, 2022

Title: Fractional revival on graphs.

Abstract: Let $A$ be the adjacency matrix of a weighted graph, and let $U(t)=e^{itA}$. If there is a time $t$ such that $U(t)e_a=\alpha e_a+\beta e_b$, then we say there is fractional revival (FR) between $a$ and $b$. For the special case when $\alpha=0$, we say there is perfect state transfer (PST) between vertices $a$ and $b$. It is known that PST is monogamous (PST from $a$ to $b$ and PST from $a$ to $c$ implies $b=c$) and vertices $a, b$ are cospectral in this case. If $\alpha\beta\neq 0$, then there is proper fractional revival. It is proven that in this case the two vertices are fractionally cospectral (a generalization of cospectrality). We will look at a family of unweighted graphs where proper FR occurs between non-cospectral vertices (no such examples were known before), and graphs with overlapping FR pairs (three different vertices $a$, $b$, and $c$, with FR between $a,b$, and FR between $a,c$).

You can watch the talk here, and the slides are available here.

This talk was given on November 16th, 2020