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.