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ELA

THE MINIMUM RANK PROBLEM OVER FINITE FIELDS

∗

JASON GROUT†

Abstract. The structure of all graphs having minimum rank at mostkover a finite field withq

elements is characterized for any possiblekandq. A strong connection between this characterization and polarities of projective geometries is explained. Using this connection, a few results in the minimum rank problem are derived by applying some known results from projective geometry.

Key words. Minimum rank, Symmetric matrix, Finite field, Projective geometry, Polarity graph, Bilinear symmetric form

AMS subject classifications. 05C50, 05C75, 15A03, 05B25, 51E20.

∗Received by the editors January 17, 2008. Accepted for publication July 13, 2010. Handling

Editor: Robert Guralnick.

†Department of Mathematics and Computer Science, Drake University, 2507 University Avenue,

Des Moines, IA 50311, USA ([email protected]).

Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 20, pp. 691-716, November 2010

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