Zero forcing sets and the minimum rank of graphs

The minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i not equal j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. This paper introduces a new graph parameter, Z(G), that is the minimum size of a zero forcing set of vertices and uses it to bound the minimum rank for numerous families of graphs, often enabling computation of the minimum rank. (c) 2007 Elsevier Inc. All rights reserved.