Signless Laplacian spectral determinations of some multicone graphs

If a clique and a regular graph are joined together the resulting graph is called a multicone graph. A graph G is said to be determined by the spectrum of its signless Laplacian matrix (DQS, for short) if every graph with the same Laplacian spectrum is isomorphic to G. It is proved that all the multicone graphs Kr ▽ sKt, except for Kr ▽ 3K1, are DQS, where Kr denots a complete graph with r ≥ 1 vertices. Consequently, by using these results we give a response to an open problem in [24].