Zero forcing sets and the minimum rank of graphs

Abstract

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 ≠ 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.