Mathematics – Combinatorics
Scientific paper
2010-03-10
Mathematics
Combinatorics
14 pages, 2 figures. To appear in Linear Algebra and its Applications.
Scientific paper
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.
Barioli Francesco
Barrett Wayne
den Driessche van P.
der Holst Hein van
Fallat Shaun M.
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