Universally optimal matrices and field independence of the minimum rank of a graph

Date
2009-07-01
Authors
DeAlba, Luz
Grout, Jason
Hogben, Leslie
Hogben, Leslie
Mikkelson, Rana
Rasmussen, Kaela
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Altmetrics
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Research Projects
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Mathematics
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Abstract

The minimum rank of a simple graph G over a field F is the smallest possible rank among all symmetric matrices over F whose (i, j)th entry ( for i not equal j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. A universally optimal matrix is defined to be an integer matrix A such that every off-diagonal entry of A is 0, 1, or -1, and for all fields F, the rank of A is the minimum rank over F of its graph. Universally optimal matrices are used to establish field independence of minimum rank for numerous graphs. Examples are also provided verifying lack of field independence for other graphs.

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<p>This article is published as DeAlba, Luz, Jason Grout, Leslie Hogben, Rana Mikkelson, and Kaela Rasmussen. "Universally optimal matrices and field independence of the minimum rank of a graph." <em>The Electronic Journal of Linear Algebra</em> 18 (2009): 403-419. DOI: <a href="https://doi.org/10.13001/1081-3810.1321" target="_blank">10.13001/1081-3810.1321</a>. Posted with permission.</p>
Keywords
Minimum rank, Universally optimal matrix, Field independent, Symmetric matrix, Rank, Graph, Matrix
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