TY - JOUR

T1 - Zero forcing parameters and minimum rank problems

AU - Barioli, F.

AU - Barrett, W.

AU - Fallat, S.M.

AU - Hall, H.T.

AU - Hogben, L.

AU - Shader, B.L.

AU - Driessche, van den, P.

AU - Holst, van der, H.

PY - 2010

Y1 - 2010

N2 - The zero forcing number Z(G), which is the minimum number of vertices in a zero forcing set of a graph G, is used to study the maximum nullity/minimum rank of the family of symmetric matrices described by G. It is shown that for a connected graph of order at least two, no vertex is in every zero forcing set. The positive semidefinite zero forcing number Z+(G) is introduced, and shown to be equal to |G|-OS(G), where OS(G) is the recently defined ordered set number that is a lower bound for minimum positive semidefinite rank. The positive semidefinite zero forcing number is applied to the computation of positive semidefinite minimum rank of certain graphs. An example of a graph for which the real positive symmetric semidefinite minimum rank is greater than the complex Hermitian positive semidefinite minimum rank is presented.

AB - The zero forcing number Z(G), which is the minimum number of vertices in a zero forcing set of a graph G, is used to study the maximum nullity/minimum rank of the family of symmetric matrices described by G. It is shown that for a connected graph of order at least two, no vertex is in every zero forcing set. The positive semidefinite zero forcing number Z+(G) is introduced, and shown to be equal to |G|-OS(G), where OS(G) is the recently defined ordered set number that is a lower bound for minimum positive semidefinite rank. The positive semidefinite zero forcing number is applied to the computation of positive semidefinite minimum rank of certain graphs. An example of a graph for which the real positive symmetric semidefinite minimum rank is greater than the complex Hermitian positive semidefinite minimum rank is presented.

U2 - 10.1016/j.laa.2010.03.008

DO - 10.1016/j.laa.2010.03.008

M3 - Article

SN - 0024-3795

VL - 433

SP - 401

EP - 411

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - 2

ER -