Geodesic
Universally optimal matrices and field independence of the minimum rank of a graph
The electronic journal of linear algebra, Tome 18 (2009), pp. 403-419.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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 = j) isnonzero whenever ${i, j}$ isan 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 is0, 1, or - 1, and for all fields F , the rank of A isthe minimum rank over F of its graph. Universally optimal matrices are used to establish field independence of minimum rank for numerousgraphs. Examplesare also provided verifying lack of field independence for other graphs.
Classification : 05C50, 15A03
Keywords: minimum rank, universally optimal matrix, field independent, symmetric matrix, rank, graph, matrix 
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     title = {Universally optimal matrices and field independence of the minimum rank of a graph},
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Dealba, Luz M.; Grout, Jason; Hogben, Leslie; Mikkelson, Rana; Rasmussen, Kaela. Universally optimal matrices and field independence of the minimum rank of a graph. The electronic journal of linear algebra, Tome 18 (2009), pp. 403-419. http://geodesic.mathdoc.fr/item/ELA_2009__18__a25/