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Vertex and edge spread of zero forcing number, maximum nullity, and minimum rank of a graph. (English) Zbl 1241.05076
Summary: The minimum rank of a simple graph $$G$$ is defined to be the smallest possible rank over all symmetric real matrices whose $$ij$$th entry (for $$i\neq j$$) is nonzero whenever $$\{i,j\}$$ is an edge in $$G$$ and is zero otherwise; maximum nullity is taken over the same set of matrices.
The zero forcing number is the minimum size of a zero forcing set of vertices and bounds the maximum nullity from above. The spread of a graph parameter at a vertex $$v$$ or edge $$e$$ of $$G$$ is the difference between the value of the parameter on $$G$$ and on $$G-v$$ or $$G-e$$. Rank spread (at a vertex) was introduced in [F. Barioli, S. M. Fallar and I. Hogben, Linear Algebra Appl. 392, 289–303 (2004; Zbl 1052.05045)].
This paper introduces vertex spread of the zero forcing number and edge spreads for minimum rank/maximum nullity and zero forcing number. Properties of the spreads are established and used to determine values of the minimum rank/maximum nullity and zero forcing number for various types of grids with a vertex or edge deleted.

##### MSC:
 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 15A03 Vector spaces, linear dependence, rank, lineability 15A18 Eigenvalues, singular values, and eigenvectors
Zbl 1052.05045
Full Text:
##### References:
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