$Q$-spectrum; $Q$-controllable graph; $Q$-cospectral graph.
A connected graph is called Q-controllable if its signless Laplacian eigenvalues are mutually distinct and main. Two graphs G and H are said to be Q-cospectral if they share the same signless Laplacian spectrum. In this paper, infinite families of Q-controllable graphs are constructed, by using the operator of rooted product introduced by Godsil and McKay. In the process, innitely many non-isomorphic Q-cospectral graphs are also constructed, especially, including those graphs whose signless Laplacian eigenvalues are mutually distinct.
Lou, Zhenzhen; Huang, Qiongxiang; and Huang, Xueyi.
"On the construction of Q-controllable graphs",
Electronic Journal of Linear Algebra,
Volume 32, pp. 365-379.