The maximum nullity of a complete subdivision graph is equal to its zero forcing number

Date
2014-06-01
Authors
Barrett, Wayne
Butler, Steve
Catral, Minerva
Hogben, Leslie
Fallat, Shaun
Hall, H. Tracy
Hogben, Leslie
Young, Michael
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Abstract

Barrett et al. asked in [W. Barrett et al. Minimum rank of edge subdivisions of graphs. Electronic Journal of Linear Algebra, 18:530-563, 2009.], whether the maximum nullity is equal to the zero forcing number for all complete subdivision graphs. We prove that this equality holds. Furthermore, we compute the value of M(F,(G) = Z(G) by introducing the bridge tree of a connected graph. Since this equality is valid for all fields, G has field independent minimum rank, and we also show that G has a universally optimal matrix.

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<p>This article is published as Barrett, Wayne, Steve Butler, Minerva Catral, Shaun Fallat, H. Hall, Leslie Hogben, and Michael Young. "The maximum nullity of a complete subdivision graph is equal to its zero forcing number." <em>The Electronic Journal of Linear Algebra</em> 27 (2014): 444-457. DOI: <a href="https://doi.org/10.13001/1081-3810.1629" target="_blank">10.13001/1081-3810.1629</a>. Posted with permission.</p>
Keywords
Zero forcing number, Maximum nullity, Minimum rank, Complete subdivision, Bridge tree, Universally optimal, Matrix, Graph
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