Quantum state transfer on Q-graphs

Given a graph G, the graph Q(G) is obtained by inserting a new vertex into every edge of G, then connecting those inserted vertices on the neighbouring edges in G. We usually refer to Q(G) as the Q-graph of G. This article mainly focuses on the effects of Q-graph operation on quantum state transfer. It is shown that if G is an integral r-regular graph for r >= 2, then Q(G) always avoids perfect state transfer. In contrast, we also show that, this Q-graph exists pretty good state transfer under some special circumstances. Subsequently, by applying the results obtained above, we also display many new classes of Q-graphs that avoid perfect state transfer but exist pretty good state transfer.