Complete split graph determined by its (signless) Laplacian spectrum

A complete split graph CS(n,α), is a graph on nn vertices consisting of a clique on n−αn−α vertices and an independent set on the remaining α(1≤α≤n−1) vertices in which each vertex of the clique is adjacent to each vertex of the independent set. In this paper, we prove that CS(n,α) is determined by its Laplacian spectrum when 1≤α≤n−11≤α≤n−1, and CS(n,α) is also determined by its signless Laplacian spectrum when 1≤α≤n−11≤α≤n−1 and α≠3α≠3.