 
Summary: Linear Algebra and its Applications 426 (2007) 558582
www.elsevier.com/locate/laa
The minimum rank of symmetric matrices described
by a graph: A survey
Shaun M. Fallat a,1, Leslie Hogben b,,2
a Department of Mathematics and Statistics, University of Regina, Regina, SK, Canada
b Department of Mathematics, Iowa State University, Ames, IA 50011, USA
Received 11 January 2007; accepted 22 May 2007
Available online 15 June 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 surveys the current state of knowledge on the problem of determining the minimum rank of a
graph and related issues.
© 2007 Elsevier Inc. All rights reserved.
AMS classification: 05C50; 15A03; 15A18
Keywords: Minimum rank; Inverse eigenvalue problem; Rank; Graph; Symmetric matrix; Matrix
1. Introduction
The minimum rank problem for a simple graph (the minimum rank problem for short) asks us to
