On constructing graphs with a prescribed adjacency property

A graph G is said to have property P(m,n,k) if for any disjoint sets A and B of vertices of G with IAI = m and IBI = n there exist at least k other vertices, each of which is adjacent to every vertex of A but not adjacent to any vertex of B. We know that almost all graphs have property P(m,n.k). However, almost no graphs have been constructed. In this paper. we construct classes of graphs having property P(l,n,k). For the case m, n ~ 2, the problem of constructing graphs with the property P(m,n,k) seems difficult, with the only known examples being Paley graphs.