The rank of random graphs

Abstract

We show that almost surely the rank of the adjacency matrix of the Erdos-Rényi random graph G(n,p) equals the number of nonisolated vertices for any c In n/n < p < 1/2, where c is an arbitrary positive constant larger than 1/2. In particular, the adjacency matrix of the giant component (a.s.) has full rank in this range. © 2008 Wiley Periodicals, Inc.