Algebraic Graph Theory Seminar - Xiaohong Zhang

Monday, November 16, 2020 11:30 am - 11:30 am EST (GMT -05:00)

Title: Fractional revival on graphs

Speaker: Xiaohong Zhang
Affiliation: University of Waterloo
Zoom: Contact Soffia Arnadottir


Let A be the adjacency matrix of a weighted graph, and let U(t)=exp(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).