Summary: Linear Algebra and its Applications 428 (2008) 1628­1648
Available online at www.sciencedirect.com
www.elsevier.com/locate/laa
Zero forcing sets and the minimum rank of graphs
AIM Minimum Rank ­ Special Graphs Work Group ,1
American Institute of Mathematics 360 Portage Ave. Palo Alto, CA 94306, USA
Received 6 March 2007; accepted 8 October 2007
Available online 26 November 2007
Submitted by R.A. Brualdi
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.
© 2007 Elsevier Inc. All rights reserved.
AMS classification: 05C50; 15A18; 15A03
Keywords: Minimum rank; Rank; Graph; Symmetric matrix; Matrix
The workshop "Spectra of Families of Matrices described by Graphs, Digraphs, and Sign Patterns", was held at the
American Institute of Mathematics, October 23­27, 2006. The authors thank AIM and NSF for their support.

Source: Argerami, Martin - Department of Mathematics and Statistics, University of Regina

Collections: Mathematics