Signless Laplacian state transfer on Q-graphs

摘要

For a simple graph G, its Q-graph Q(G) is derived from G by adding one new point in every edge of G and linking two new vertices by edge if they are between two edges that having a common endpoint. In our work, we demonstrate that for a regular graph G, if all the signless Laplacian eigenvalues are integers, then the Q(G) exists no signless Laplacian perfect state transfer. We also present a sufficient restriction that the Q(G) admits signless Laplacian pretty good state transfer when G exhibits signless Laplacian perfect state transfer between two specific vertices for a regular graph G. In addition, in view of these results, we also present some new families of Q-graphs, which have no signless Laplacian perfect state transfer, but admit signless Laplacian pretty good state transfer.