Rank of adjacency matrices of directed (strongly) regular graphs

For a positive integer r we consider the set B-r of all values of k/n for which there exists an n x n matrix with entries 0 and 1 such that each row and each column has exactly k 1's and the matrix has rank r. We prove that the set B-r is finite, for every r. If there exists a k-regular directed graph on n vertices such that its adjacency matrix has rank r then k/n epsilon B-r. We use this to exclude existence of directed strongly regular graphs for infinitely many feasible parameter sets. (c) 2005 Elsevier Inc. All rights reserved.