Zero forcing parameters and minimum rank problems

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Abstract

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. © 2010 Elsevier Inc. All rights reserved.

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Barioli, F., Barrett, W., Fallat, S. M., Hall, H. T., Hogben, L., Shader, B., … van der Holst, H. (2010). Zero forcing parameters and minimum rank problems. Linear Algebra and Its Applications, 433(2), 401–411. https://doi.org/10.1016/j.laa.2010.03.008

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