๐”– Bobbio Scriptorium
โœฆ   LIBER   โœฆ

A note on universally optimal matrices and field independence of the minimum rank of a graph

โœ Scribed by Liang-Hao Huang; Gerard J. Chang; Hong-Gwa Yeh

Publisher
Elsevier Science
Year
2010
Tongue
English
Weight
164 KB
Volume
433
Category
Article
ISSN
0024-3795

No coin nor oath required. For personal study only.

โœฆ Synopsis

For a simple graph G on n vertices, the minimum rank of G over a field F, written as mr F (G), is defined to be the smallest possible rank among all n ร— n symmetric matrices over F whose (i, j)th entry (for i / = j) is nonzero whenever {i, j} is an edge in G and is zero otherwise.

A symmetric integer matrix A such that every off-diagonal entry is 0, 1, or -1 is called a universally optimal matrix if, for all fields F, the rank of A over F is the minimum rank of the graph of A over F.

๐Ÿ“œ SIMILAR VOLUMES

Minimum rank of skew-symmetric matrices
Minimum rank of skew-symmetric matrices described by a graph
โœ IMA-ISU research group on minimum rank ๐Ÿ“‚ Article ๐Ÿ“… 2010 ๐Ÿ› Elsevier Science ๐ŸŒ English โš– 337 KB

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 exte

The minimum rank of symmetric matrices d
The minimum rank of symmetric matrices described by a graph: A survey
โœ Shaun M. Fallat; Leslie Hogben ๐Ÿ“‚ Article ๐Ÿ“… 2007 ๐Ÿ› Elsevier Science ๐ŸŒ English โš– 331 KB

The minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i / = j ) is nonzero whenever {i, j } is an edge in G and is zero otherwise. This paper surveys the current state of knowledge on the problem of determining the min