Minimum rank of skew-symmetric matrices described by a graph

Date
2010-05-01
Authors
Allison, Mary
Bodine, Elizabeth
DeAlba, Luz Maria
Debnath, Joyati
DeLoss, Laura
Garnett, Colin
Grout, Jason
Hogben, Leslie
Im, Bokhee
Kim, Hana
Nair, Reshmi
Pryporova, Olga
Savage, Kendrick
Shader, Bryan
Wangsness Wehe, Amy
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Abstract

The minimum (symmetric) rank of a simple graph G over a field F is the smallest possible rank among all symmetric matrices over F whose ijth entry (for i≠j) is nonzero whenever {i,j} is an edge in G and is zero otherwise. The problem of determining minimum (symmetric) rank has been studied extensively. We define the minimum skew rank of a simple graph G to be the smallest possible rank among all skew-symmetric matrices over F whose ijth entry (for i≠j) is nonzero whenever {i,j} is an edge in G and is zero otherwise. We apply techniques from the minimum (symmetric) rank problem and from skew-symmetric matrices to obtain results about the minimum skew rank problem.

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This is a manuscript of an article from Linear Algebra and its Applications 432 (2010): 2457, doi:10.1016/j.laa.2009.10.001. Posted with permission.

Keywords
minimum rank, minimum skew rank, skew-symmetric matrix, matching, pfaffian, rank, graph, matrix
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