State transfer and star complements in graphs

Let GG be a graph with adjacency matrix AA, and let H(t)=exp(itA)H(t)=exp(itA). For an eigenvalue μμ of AA with multiplicity kk, a star set   for μμ in GG is a vertex set XX of GG such that |X|=k|X|=k and the induced subgraph G−XG−X does not have μμ as an eigenvalue. GG is said to have perfect state transfer from the vertex uu to the vertex vv if there is a time ττ such that |H(τ)u,v|=1|H(τ)u,v|=1. The unitary operator H(t)H(t) has important applications in the transfer of quantum information. In this paper, we give an expression of H(t)H(t). For a star set XX of graph GG, perfect state transfer does not occur between any two vertices in XX. We also give some results for the existence of perfect state transfer in a graph.