Distance Integral Complete r-Partite Graphs

Let D(G) = (d(ij))(nxn) denote the distance matrix of a connected graph G with order n, where dij is equal to the distance between vertices v(i) and v(j) 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,K-p2,K-...,K-pr = K-a1.p1,K-a2.p2,K-...,K-a epsilon.ps and give a su ffi cient and necessary condition for K-a1.p1,K-a2.p2,K-...,K-a epsilon.ps to be distance integral, from which we construct infinitely many new classes of distance integral graphs with s = 1, 2, 3, 4. Finally, we propose two basic open problems for further study.