Integral complete multipartite graphs Ka1⋅p1,a2⋅p2,…,as⋅psKa1⋅p1,a2⋅p2,…,as⋅ps with s=5,6s=5,6

A graph is called integral if all the eigenvalues of its adjacency matrix are integers. In our recent work, we have studied integral complete rr-partite graphs Kp1,p2,…,pr=Ka1⋅p1,a2⋅p2,…,as⋅psKp1,p2,…,pr=Ka1⋅p1,a2⋅p2,…,as⋅ps with s=3,4s=3,4 (also see, L.G. Wang, X.D. Liu, Integral complete multipartite graphs, Discrete Math. 308 (2008) 3860–3870 ). In this paper, we continue the work on such integral graphs, we investigate integral complete multipartite graphs Ka1⋅p1,a2⋅p2,…,as⋅psKa1⋅p1,a2⋅p2,…,as⋅ps with s=5,6s=5,6 for the first time by computer search. Then we construct infinite many new classes of such integral graphs by solving some certain Diophantine equations. These results are different from those in the existing literature. For s=5,6s=5,6, we give a positive answer to a question of Wang et al. [L.G. Wang, X.L. Li, C. Hoede, Integral complete rr-partite graphs, Discrete Math. 283 (2004) 231–241]. The problem of the existence of integral complete multipartite graphs Ka1⋅p1,a2⋅p2,…,as⋅psKa1⋅p1,a2⋅p2,…,as⋅ps with arbitrarily large number ss remains open.