A graph-theoretic result for a model of neural computation

We deal in this work with the following graph construction problem that arises in a model of neural computation introduced by L.G. Valiant. For an undirected graph G = (V, E), let set N∗(X, Y), where X, Y ⊂- V, denote the set of vertices other than those of X, Y which are adjacent to at least one vertex in X and at least one vertex in Y. An undirected graph G needs to be constructed that exhibits the following connectivity property. For any constant k and all disjoint sets A, B ⊂- V such that ¦A¦ = ¦B¦ = k, it is required that the cardinality of set C = N∗(A, B) be K or as close to k as possible.