Abstract:  Let G be a simple graph. The matrix Q(G)=D(G)+A(G) denotes the signless Laplacian matrix of G, where D(G) and A(G) denote the diagonal matrix and the adjacency matrix of G respectively. A graph is called Q-integral if its signless Laplacian spectrum consists entirely of integers. In this paper, we firstly construct six infinite classes of nonregular Q-integral graphs from the known six smaller Q-integral graphs identified by Stani\'{c}.  Furthermore, we obtain large families of Q-integral graphs by  the Cartesian product of graphs. Finally, we obtain some regular Q-integral graphs.

Key words:  signless Laplacian spectrum, Q-integral graph, integral graph, integral eigenvalues