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.

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.

Copyright Owner

The Author(s)

Language

en

File Format

application/pdf

Included in

Algebra Commons

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