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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.

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: DOI
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