Ela the Maximum Nullity of a Complete Subdivision Graph Is Equal to Its Zero Forcing Number∗
نویسندگان
چکیده
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, G̊) = Z(G̊) by introducing the bridge tree of a connected graph. Since this equality is valid for all fields, G̊ has field independent minimum rank, and we also show that G̊ has a universally optimal matrix.
منابع مشابه
Ela Positive Semidefinite Maximum Nullity and Zero Forcing Number
The zero forcing number Z(G) is used to study the minimum rank/maximum nullity of the family of symmetric matrices described by a simple, undirected graph G. The positive semidefinite zero forcing number is a variant of the (standard) zero forcing number, which uses the same definition except with a different color-change rule. The positive semidefinite maximum nullity and zero forcing number f...
متن کاملEla on the Minimum Rank of Not Necessarily Symmetric Matrices: a Preliminary Study∗
The minimum rank of a directed graph Γ is defined to be the smallest possible rank over all real matrices whose ijth entry is nonzero whenever (i, j) is an arc in Γ and is zero otherwise. The symmetric minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i = j) is nonzero whenever {i, j} is an edge in G and is zero o...
متن کاملEla Minimum Rank, Maximum Nullity, and Zero Forcing Number of Simple Digraphs
A simple digraph describes the off-diagonal zero-nonzero pattern of a family of (not necessarily symmetric) matrices. Minimum rank of a simple digraph is the minimum rank of this family of matrices; maximum nullity is defined analogously. The simple digraph zero forcing number is an upper bound for maximum nullity. Cut-vertex reduction formulas for minimum rank and zero forcing number for simpl...
متن کاملEla Note on Positive Semidefinite Maximum Nullity and Positive Semidefinite Zero Forcing Number of Partial 2-trees
The maximum positive semidefinite nullity of a multigraph G is the largest possible nullity over all real positive semidefinite matrices whose (i, j)th entry (for i 6= j) is zero if i and j are not adjacent in G, is nonzero if {i, j} is a single edge, and is any real number if {i, j} is a multiple edge. The definition of the positive semidefinite zero forcing number for simple graphs is extende...
متن کاملPositive semidefinite maximum nullity and zero forcing number
The zero forcing number Z(G) is used to study the minimum rank/maximum nullity of the family of symmetric matrices described by a simple, undirected graph G. The positive semidefinite zero forcing number is a variant of the (standard) zero forcing number, which uses the same definition except with a different color-change rule. The positive semidefinite maximum nullity and zero forcing number f...
متن کامل