The maximum nullity of a complete subdivision graph is equal to its zero forcing number
Wayne W. Barrett, Steve Butler, Minerva Catral, Shaun Fallat, H. Tracy Hall, Leslie Hogben, Michael Young · Electronic Journal of Linear Algebra · 2014
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.