Some graphs determined by their signless laplacian (Distance) spectra

In literature, there are some results known about spectral determination of graphs with many edges. In [M. C´amara and W.H. Haemers. Spectral characterizations of almost complete graphs. Discrete Appl. Math., 176:19–23, 2014.], Ca´mara and Haemers studied complete graph with some edges deleted for spectral determination. In fact, they found that if the deleted edges form a matching, a complete graph Km provided m ≤ n−2, or a complete bipartite graph, then it is determined by its adjacency spectrum. In this paper, the graph Kn\Kl,m (n > l + m) which is obtained from the complete graph Kn by removing all the edges of a complete bipartite subgraph Kl,m is studied. It is shown that the graph Kn\K1,m with m ≥ 4 is determined by its signless Laplacian spectrum, and it is proved that the graph Kn\Kl,m is determined by its distance spectrum. The signless Laplacian spectral determination of the multicone graph Kn−2α ∨ αK2 was studied by Bu and Zhou in [C. Bu and J. Zhou. Signless Laplacian spectral characterization of the cones over some regular graphs. Linear Algebra Appl., 436:3634–3641, 2012.] and Xu and He in [L. Xu and C. He. On the signless Laplacian spectral determination of the join of regular graphs. Discrete Math. Algorithm. Appl., 6:1450050, 2014.] only for n−2α = 1 or 2. Here, this problem is completely solved for all positive integer n−2α. The proposed approach is entirely different from those given by Bu and Zhou, and Xu and He.