On the minimum rank of not necessarily symmetric matrices: A preliminary study

Date
2009-02-01
Authors
Barioli, Francesco
Fallat, Shaun
Hall, H. Tracy
Hershkowitz, Daniel
Hogben, Leslie
van der Holst, Hein
Shader, Bryan
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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.

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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.

Keywords
Minimum rank, Maximum nullity, symmetric minimum rank, Asymmetric minimum rank, Path cover number, Zero forcing set, Zero forcing number, Edit distance, Triangle number, Minimum degree, Ditree, Directed tree, Inverse eigenvalue problem, Rank, Graph, Symmetric matrix
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