Campus Units
Mathematics
Document Type
Article
Publication Version
Published Version
Publication Date
2-2009
Journal or Book Title
Electronic Journal of Linear Algebra
Volume
18
First Page
126
Last Page
145
DOI
10.13001/1081-3810.1300
Abstract
The minimum rank of a directed graph Γ is defined to be the smallest possible rank over all real matrices whose ijth entry is nonzero whenever (i,j) is an arc in Γ and is zero otherwise. The symmetric 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. Maximum nullity is equal to the difference between the order of the graph and minimum rank in either case. Definitions of various graph parameters used to bound symmetric maximum nullity, including path cover number and zero forcing number, are extended to digraphs, and additional parameters related to minimum rank are introduced. It is shown that for directed trees, maximum nullity, path cover number, and zero forcing number are equal, providing a method to compute minimum rank for directed trees. It is shown that the minimum rank problem for any given digraph or zero-nonzero pattern may be converted into a symmetric minimum rank problem.
Copyright Owner
The Author(s)
Copyright Date
2009
Language
en
File Format
application/pdf
Recommended Citation
Barioli, Francesco; Fallat, Shaun M.; Hall, H. Tracy; Hershkowitz, Daniel; Hogben, Leslie; van der Holst, Hein; and Shader, Bryan, "On the minimum rank of not necessarily symmetric matrices: A preliminary study" (2009). Mathematics Publications. 242.
https://lib.dr.iastate.edu/math_pubs/242
Comments
This article is published as Barioli, Francesco, Shaun Fallat, H. Hall, Daniel Hershkowitz, Leslie Hogben, Hein Van der Holst, and Bryan Shader. "On the minimum rank of not necessarily symmetric matrices: a preliminary study." The Electronic Journal of Linear Algebra 18 (2009): 126-145. DOI: 10.13001/1081-3810.1300. Posted with permission.