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. We prove that for k > 1, if any graph G exists so that for all choices of A and B, set C = N* (A, B) has cardinality exactly k, then G must have exactly 3k vertices. Thus an exact solution for arbitrarily large values of n does not exist for any such k. A graph construction based on a projective plane graph provides an approximate solution, in a certain sense, for arbitrarily large n.