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 $a,c$).