Mathematics, Electrical and Computer Engineering
Journal or Book Title
Electronic Journal of Linear Algebra
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.
Barrett, Wayne; Butler, Steve; Catral, Minerva; Fallat, Shaun M.; Hall, H. Tracy; Hogben, Leslie; and Young, Michael, "The maximum nullity of a complete subdivision graph is equal to its zero forcing number" (2014). Mathematics Publications. 229.