Signless Laplacian state transfer on Q -graphs

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. (c) 2022 Elsevier Inc. All rights reserved.