Q-integral complete r-partite graphs

For a graph G of order n, the signless Laplacian matrix of G is Q(G)=D(G)+A(G), where A(G) is its adjacency matrix and D(G) is the diagonal matrix of the vertex degrees in G. The signless Laplacian characteristic polynomial (or Q-polynomial) of G is Q_G(x)=|x I_n-Q(G)|, where I_n is the n×n identity matrix. A graph G is called Q-integral if all the eigenvalues of its signless Laplacian characteristic polynomial Q_G(x) are integers. In this paper, we give a sufficient and necessary condition for complete r-partite graphs to be Q-integral, from which we construct infinitely many new classes of Q-integral graphs. Finally, we propose two basic open problems for further study.