Abstract: For a simple graph G, let matrix Q(G)=D(G) + A(G) be it’s signless Laplacian matrix and Q_G(λ)=det(λI Q) it’s signless Laplacian characteristic polynomial, where D(G) denotes the diagonal matrix of vertex degrees of G, A(G) denotes its adjacency matrix of G. If all eigenvalues of Q_G(λ) are integral, then the graph G is called Q-integral. In this paper, we obtain that the signless Laplacian characteristic polynomials of the complete multi-partite graphs G=K_{n1,n2,···,nt}. We prove that the complete t-partite graphs K_{(n,n,···,n)}t are Q-integral and give a necessary and sufficient condition for the complete multipartite graphs K_{(m,···,m)s(n,···,n)}t to be Q-integral. We also obtain that the signless Laplacian characteristic polynomials of the complete multipartite graphs K_{(m,···,m,)s1(n,···,n,)s2(l,···,l)s3}. 

Key words: the signless Laplacian spectrum, the complete multipartite graphs, the Qintegral