Computational and Theoretical Challenges for Computing the Minimum Rank of a Graph

被引：3

作者：

Hicks, Illya, V ^{[1
]}

Brimkov, Boris ^{[2
]}

Deaett, Louis ^{[3
]}

Haas, Ruth ^{[4
]}

Mikesell, Derek ^{[1
]}

Roberson, David ^{[5
,6
]}

Smith, Logan ^{[1
]}

机构：

[1] Rice Univ, Computat Appl Math & Operat Res Dept, Houston, TX 77005 USA

[2] Slippery Rock Univ, Math & Stat Dept, Slippery Rock, PA 16057 USA

[3] Quninnipiac Univ, Math & Stat Dept, Hamden, CT 06518 USA

[4] Univ Hawaii Monoa, Dept Math, Honolulu, HI 96822 USA

[5] Tech Univ Denmark, Dept Appl Math & Comp Sci, DK-2800 Lyngby, Denmark

[6] Univ Copenhagen, Ctr Math Quantum Theory QMATH, Dept Math Sci, DK-2100 Copenhagen, Denmark

来源：

INFORMS JOURNAL ON COMPUTING | 2022年 / 34卷 / 06期

基金：

美国国家科学基金会;

关键词：

minimum rank; maximum nullity; matroid; zero forcing; forts; ZERO FORCING SETS; PATH COVER NUMBER; MAXIMUM NULLITY; PARAMETERS;

D O I：

10.1287/ijoc.2022.1219

中图分类号：

TP39 [计算机的应用];

学科分类号：

081203 ; 0835 ;

摘要：

The minimum rank of a graph G is the minimum of the ranks of all symmetric adjacency matrices of G. We present a new combinatorial bound for the minimum rank of an arbitrary graph G based on enumerating certain subsets of vertices of G satisfying matroid theoretic properties. We also present some computational and theoretical challenges associated with computing the minimum rank. This includes a conjecture that this bound on the minimum rank actually holds with equality for all graphs.

引用

页码：2868 / 2872

页数：5

共 50 条

[1] An upper bound for the minimum rank of a graph
Berman, Avi
Friedland, Shmuel
Hogben, Leslie
Rothblum, Uriel G.
Shader, Bryan
[J]. LINEAR ALGEBRA AND ITS APPLICATIONS, 2008, 429 (07) : 1629 - 1638
[2] On the graph complement conjecture for minimum rank
Barioli, Francesco
Barrett, Wayne
Fallat, Shaun M.
Hall, H. Tracy
Hogben, Leslie
van der Holst, Hein
[J]. LINEAR ALGEBRA AND ITS APPLICATIONS, 2012, 436 (12) : 4373 - 4391
[3] On the minimum semidefinite rank of a simple graph
Booth, Matthew
Hackney, Philip
Harris, Benjamin
Johnson, Charles R.
Lay, Margaret
Lenker, Terry D.
Mitchell, Lon H.
Narayan, Sivaram K.
Pascoe, Amanda
Sutton, Brian D.
[J]. LINEAR & MULTILINEAR ALGEBRA, 2011, 59 (05): : 483 - 506
[4] On the graph complement conjecture for minimum semidefinite rank
Mitchell, Lon H.
[J]. LINEAR ALGEBRA AND ITS APPLICATIONS, 2011, 435 (06) : 1311 - 1314
[5] Techniques for determining the minimum rank of a small graph
DeLoss, Laura
Grout, Jason
Hogben, Leslie
McKay, Tracy
Smith, Jason
Tims, Geoff
[J]. LINEAR ALGEBRA AND ITS APPLICATIONS, 2010, 432 (11) : 2995 - 3001
[6] Minimum-rank matrices with prescribed graph
[J]. Linear Algebra Its Appl, (303):
[7] On minimum rank and zero forcing sets of a graph
Huang, Liang-Hao
Chang, Gerard J.
Yeh, Hong-Gwa
[J]. LINEAR ALGEBRA AND ITS APPLICATIONS, 2010, 432 (11) : 2961 - 2973
[8] On the minimum rank of adjacency matrices of regular graph
Liang, Xiu-dong
[J]. Advances in Matrix Theory and Applications, 2006, : 346 - 348
[9] On the minimum rank of a graph over finite fields
Friedland, Shmuel
Loewy, Raphael
[J]. LINEAR ALGEBRA AND ITS APPLICATIONS, 2012, 436 (06) : 1710 - 1720
[10] Minimum-rank matrices with prescribed graph
Nylen, PM
[J]. LINEAR ALGEBRA AND ITS APPLICATIONS, 1996, 248 : 303 - 316

← 1 2 3 4 5 →