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\'{a}mara and W.H.~Haemers. Spectral characterizations of almost complete graphs. {\em Discrete Appl. Math.}, 176:19--23, 2014.], C\'amara 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 $K_m$ provided $m \le n-2$, or a complete bipartite graph, then it is determined by its adjacency spectrum. In this paper, the graph $K_{n}\backslash K_{l,m}$ $(n>l+m)$ which is obtained from the complete graph $K_{n}$ by removing all the edges of a complete bipartite subgraph $K_{l,m}$ is studied. It is shown that the graph $K_{n}\backslash K_{1,m}$ with $m\ge4$ is determined by its signless Laplacian spectrum, and it is proved that the graph $K_{n}\backslash K_{l,m}$ is determined by its distance spectrum. The signless Laplacian spectral determination of the multicone graph $K_{n-2\alpha}\vee \alpha K_{2}$ was studied by Bu and Zhou in [C.~Bu and J.~Zhou. Signless Laplacian spectral characterization of the cones over some regular graphs. {\em 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. {\em Discrete Math. Algorithm. Appl.}, 6:1450050, 2014.] only for $n-2\alpha=1 ~\text{or}~ 2$. Here, this problem is completely solved for all positive integer $n-2\alpha$. The proposed approach is entirely different from those given by Bu and Zhou, and Xu and He.