Signless Laplacian spectral characterization of the cones over some regular graphs

Let G be an r-regular graph of order n. We prove that the cone over G is determined by its signless Laplacian spectrum for r=1,n-2, for r=2 and n⩾11. For r=n-3, we show that the cone over G is determined by its signless Laplacian spectrum if and only if the complement of G has no triangles. A class of Q-cospectral graphs are also given.