Zero forcing parameters and minimum rank problems

F. Barioli, W. Barrett, S.M. Fallat, H.T. Hall, L. Hogben, B.L. Shader, P. Driessche, van den, H. Holst, van der

Research output: Contribution to journalArticleAcademicpeer-review

94 Citations (Scopus)

Abstract

The zero forcing number Z(G), which is the minimum number of vertices in a zero forcing set of a graph G, is used to study the maximum nullity/minimum rank of the family of symmetric matrices described by G. It is shown that for a connected graph of order at least two, no vertex is in every zero forcing set. The positive semidefinite zero forcing number Z+(G) is introduced, and shown to be equal to |G|-OS(G), where OS(G) is the recently defined ordered set number that is a lower bound for minimum positive semidefinite rank. The positive semidefinite zero forcing number is applied to the computation of positive semidefinite minimum rank of certain graphs. An example of a graph for which the real positive symmetric semidefinite minimum rank is greater than the complex Hermitian positive semidefinite minimum rank is presented.
Original languageEnglish
Pages (from-to)401-411
JournalLinear Algebra and Its Applications
Volume433
Issue number2
DOIs
Publication statusPublished - 2010

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