TY - JOUR

T1 - A note on universally optimal matrices and field independence of the minimum rank of a graph

AU - Huang, Liang Hao

AU - Chang, Gerard J.

AU - Yeh, Hong Gwa

N1 - Funding Information:
Corresponding author. E-mail addresses: r90221003@gmail.com (L.-H. Huang), gjchang@math.ntu.edu.tw (G.J. Chang), hgyeh@math.ncu.edu.tw (H.-G. Yeh). 1 Partially supported by National Science Council under Grant NSC98-2811-M-008-072. 2 Partially supported by National Science Council under Grant NSC98-2115-M-002-013-MY3. 3 Partially supported by National Science Council under Grant NSC97-2628-M-008-018-MY3.

PY - 2010/9/1

Y1 - 2010/9/1

N2 - For a simple graph G on n vertices, the minimum rank of G over a field F, written as mrF (G), is defined to be the smallest possible rank among all n × 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. Recently, Dealba et al. [L.M. Dealba, J. Grout, L. Hogben, R. Mikkelson, K. Rasmussen, Universally optimal matrices and field independence of the minimum rank of a graph, Electron. J. Linear Algebra 18 (2009) 403-419] initiated the study of universally optimal matrices and established field independence or dependence of minimum rank for some families of graphs. In the present paper, more results on universally optimal matrices and field independence or dependence of the minimum rank of a graph are presented, and some results of Dealba et al. [5] are improved.

AB - For a simple graph G on n vertices, the minimum rank of G over a field F, written as mrF (G), is defined to be the smallest possible rank among all n × 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. Recently, Dealba et al. [L.M. Dealba, J. Grout, L. Hogben, R. Mikkelson, K. Rasmussen, Universally optimal matrices and field independence of the minimum rank of a graph, Electron. J. Linear Algebra 18 (2009) 403-419] initiated the study of universally optimal matrices and established field independence or dependence of minimum rank for some families of graphs. In the present paper, more results on universally optimal matrices and field independence or dependence of the minimum rank of a graph are presented, and some results of Dealba et al. [5] are improved.

KW - Field independent

KW - Graph

KW - Matrix

KW - Maximum nullity

KW - Minimum rank

KW - Rank

KW - Symmetric matrix

KW - Universally optimal matrix

UR - http://www.scopus.com/inward/record.url?scp=77953321642&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2010.03.027

DO - 10.1016/j.laa.2010.03.027

M3 - 期刊論文

AN - SCOPUS:77953321642

SN - 0024-3795

VL - 433

SP - 585

EP - 594

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - 3

ER -