Minimum Rank Problems
By Leslie Hogben. 2010
97 citations recorded by the source corpus.
Abstract A graph describes the zero–nonzero pattern of a family of matrices, with the type of graph (undirected or directed, simple or allowing loops) determining what type of matrices (symmetric or not necessarily symmetric, diagonal entries free or constrained) are described by the graph. The minimum rank problem of the graph is to determine the minimum among the ranks of the matrices in this family; the determination of maximum nullity is equivalent. This problem has been solved for simple trees [P.M. Nylen, Minimum-rank matrices with prescribed graph, Linear Algebra Appl. 248 (1996) 303–316, C.R. Johnson, A. Leal Duarte, The maximum multiplicity of an eigenvalue in a matrix whose graph is a tree, Linear and Multilinear Algebra 46 (1999) 139–144], trees allowing loops [L.M. DeAlba, T.L. Hardy, I.R. Hentzel, L. Hogben, A. Wangsness. Minimum rank and maximum eigenvalue multiplicity of symmetric tree sign patterns, Linear Algebra Appl. 418 (2006) 389–415], and directed trees allowing loops [F. Barioli, S. Fallat, D. Hershkowitz, H.T. Hall, L. Hogben, H. van der Holst, B. Shader, On the minimum rank of not necessarily symmetric matrices: a preliminary study, Electron. J. Linear Algebra 18 (2000) 126–145]. We survey these results from a unified perspective and solve the minimum rank problem for simple directed trees.
Subfields: Computational Theory and Mathematics, Electrical and Electronic Engineering
View Paper: https://www.semanticscholar.org/paper/ebfc6da198123a2994e0a1fd6ed30aaf6a2adb5c
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In Computational Theory and Mathematics
- Darren D. Row
- Leslie Hogben
- My Huynh
- Joshua C. LaGrange
- Christina Edholm
- Shaun M. Fallat
- Samet Oymak
- Necmiye Ozay
In Electrical and Electronic Engineering
Citations in the corpus
- Vertex and edge spread of zero forcing number, maximum nullity, and minimum rank of a graph (2012, background): “The zero forcing number was introduced in [1] and the associated terminology was extended in [2, 3, 11].”