A graph-theoretic result for a model of neural computation

✍ Scribed by Alexandros V. Gerbessiotis

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 462 KB
• Volume: 82
• Category: Article
• ISSN: 0166-218X
• DOI: 10.1016/s0166-218x(97)00021-8

✦ Synopsis

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 = (I'. E). let set N*(X, Y ), where X, Y 2 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 C V such that IA\ = (BI = k, it is required that the cardinality of set C = N*(A. B) bc /, 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 U. 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 arbitrartly large II.