TY - JOUR

T1 - On the minimum rank of not necessarily symmetric matrices : a preliminary study

AU - Barioli, F.

AU - Fallat, S.M.

AU - Hall, H.T.

AU - Hershkowitz, D.

AU - Hogben, L.

AU - Holst, van der, H.

AU - Shader, B.L.

PY - 2009

Y1 - 2009

N2 - The minimum rank of a directed graph G is defined to be the smallest possible rank
over all real matrices whose ijth entry is nonzero whenever (i, j) is an arc in G 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.

AB - The minimum rank of a directed graph G is defined to be the smallest possible rank
over all real matrices whose ijth entry is nonzero whenever (i, j) is an arc in G 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.

M3 - Article

VL - 18

SP - 126

EP - 145

JO - Electronic Journal of Linear Algebra

JF - Electronic Journal of Linear Algebra

SN - 1081-3810

ER -