Algebraic Graph Theory Seminar - Xiaohong Zhang

Title: Fractional revival on graphs

Abstract:

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).