State transfer and star complements in graphs

Let G be a graph with adjacency matrix A, and let H (t) = exp(itA). For an eigenvalue mu of A with multiplicity k, a star set for mu in G is a vertex set X of G such that vertical bar X vertical bar = k and the induced subgraph G - X does not have mu as an eigenvalue. G is said to have perfect state transfer from the vertex u to the vertex v if there is a time tau such that vertical bar H(tau)(u,v)vertical bar = 1. The unitary operator H(t) has important applications in the transfer of quantum information. In this paper, we give an expression of H (t). For a star set X of graph G, perfect state transfer does not occur between any two vertices in X. We also give some results for the existence of perfect state transfer in a graph. (C) 2013 Elsevier B.V. All rights reserved.