Integral complete multipartite graphs

A graph is called integral if all eigenvalues of its adjacency matrix are integers. In this paper, we investigate integral complete r  -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. We can 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=4s=4, we give a positive answer to a question of Wang et al. [Integral complete r  -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 s remains open.