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

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Date
2014-06Author
Barrett, Wayne
Butler, Steve
Catral, Minnie
Hall, Tracy
Fallat, Shaun
Hogben, Leslie
Young, Michael
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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.