Laplacian state transfer in Q-graph

The Q-graph of a graph G is defined to be the graph obtained from G by inserting a new vertex into each edge of G, and joining by edges those pairs of new vertices which lie on adjacent edges of G. In this paper, we investigate the existence of Laplacian perfect state transfer and Laplacian pretty good state transfer in Q-graphs of r-regular graphs for r ≥ 2. We prove that there is no Laplacian perfect state transfer in the Q-graph of an r-regular graph, if r+1 is a prime number. In contrast, we give sufficient conditions for the Q-graph of an r-regular graph, where r+1 is a prime number, to have Laplacian pretty good state transfer.