Distance integral complete r-partite graphs

Let D(G) = (d_ij)_n×n denote the distance matrix of a connected graph G with order n, where d_ij is equal to the distance between vertices v_i and v_i in G. A graph is called distance integral if all eigenvalues of its distance matrix are integers. In this paper, we investigate distance integral complete r-partite graphs K_p1,p2,...,pr = K_{a1.p1,a2.p2,...,as.ps} and give a sufficient and necessary condition for K_{a1.p1,a2.p2,...,as.ps} to be distance integral, from which we construct infinitely many new classes of distance integral graphs with s = 1,2,3,4.