Zero forcing parameters and minimum rank problems
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.
Cite
@article{arxiv.1003.2028,
title = {Zero forcing parameters and minimum rank problems},
author = {Francesco Barioli and Wayne Barrett and Shaun M. Fallat and H. Tracy Hall and Leslie Hogben and Bryan Shader and P. van den Driessche and Hein van der Holst},
journal= {arXiv preprint arXiv:1003.2028},
year = {2010}
}
Comments
14 pages, 2 figures. To appear in Linear Algebra and its Applications.