A fractional approach to minimum rank and zero forcing

2015 ◽  
Author(s):  
Kevin Francis Palmowski
Keyword(s):  
Zero Forcing ◽  
Minimum Rank

scholarly journals The Bipartite Zero Forcing Set for a Full Sign Pattern Matrix

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 354
Author(s):  
Gu-Fang Mou ◽  
Tian-Fei Wang ◽  
Zhong-Shan Li
Keyword(s):  
Sign Pattern ◽  
Zero Forcing ◽  
Minimum Rank ◽  
Zero Forcing Number ◽  
Maximum Nullity ◽  
New Variant ◽  
Forcing Number ◽  
Pattern Matrix ◽  
Forcing Set ◽  
Zero Forcing Set

For an m × n sign pattern P, we define a signed bipartite graph B ( U , V ) with one set of vertices U = { 1 , 2 , … , m } based on rows of P and the other set of vertices V = { 1 ′ , 2 ′ , … , n ′ } based on columns of P. The zero forcing number is an important graph parameter that has been used to study the minimum rank problem of a matrix. In this paper, we introduce a new variant of zero forcing set−bipartite zero forcing set and provide an algorithm for computing the bipartite zero forcing number. The bipartite zero forcing number provides an upper bound for the maximum nullity of a square full sign pattern P. One advantage of the bipartite zero forcing is that it can be applied to study the minimum rank problem for a non-square full sign pattern.

scholarly journals Critical ideals, minimum rank and zero forcing number

2019 ◽  
Vol 358 ◽  
pp. 305-313 ◽  
Author(s):  
Carlos A. Alfaro ◽  
Jephian C.-H. Lin
Keyword(s):  
Zero Forcing ◽  
Minimum Rank ◽  
Zero Forcing Number ◽  
Forcing Number

scholarly journals Vertex and edge spread of zero forcing number, maximum nullity, and minimum rank of a graph

2012 ◽  
Vol 436 (12) ◽  
pp. 4352-4372 ◽  
Author(s):  
Christina J. Edholm ◽  
Leslie Hogben ◽  
My Huynh ◽  
Joshua LaGrange ◽  
Darren D. Row
Keyword(s):  
Zero Forcing ◽  
Minimum Rank ◽  
Zero Forcing Number ◽  
Maximum Nullity ◽  
Forcing Number

scholarly journals On minimum rank and zero forcing sets of a graph

2010 ◽  
Vol 432 (11) ◽  
pp. 2961-2973 ◽  
Author(s):  
Liang-Hao Huang ◽  
Gerard J. Chang ◽  
Hong-Gwa Yeh
Keyword(s):  
Zero Forcing ◽  
Minimum Rank

scholarly journals Minimum rank, maximum nullity and zero forcing number for selected graph families

2010 ◽  
Vol 3 (4) ◽  
pp. 371-392 ◽  
Author(s):  
Edgard Almodovar ◽  
Laura DeLoss ◽  
Leslie Hogben ◽  
Kirsten Hogenson ◽  
Kaitlyn Murphy ◽  
...  
Keyword(s):  
Zero Forcing ◽  
Minimum Rank ◽  
Zero Forcing Number ◽  
Maximum Nullity ◽  
Forcing Number ◽  
Graph Families

scholarly journals On zero forcing number of graphs and their complements

2015 ◽  
Vol 07 (01) ◽  
pp. 1550002 ◽  
Author(s):  
Linda Eroh ◽  
Cong X. Kang ◽  
Eunjeong Yi
Keyword(s):  
Color Change ◽  
Minimum Degree ◽  
Work Group ◽  
Black Vertex ◽  
White Vertex ◽  
Zero Forcing ◽  
Minimum Rank ◽  
Minimum Cardinality ◽  
Zero Forcing Number ◽  
Forcing Number

The zero forcing number, Z(G), of a graph G is the minimum cardinality of a set S of black vertices (whereas vertices in V(G)\S are colored white) such that V(G) is turned black after finitely many applications of "the color-change rule": a white vertex is converted to a black vertex if it is the only white neighbor of a black vertex. Zero forcing number was introduced and used to bound the minimum rank of graphs by the "AIM Minimum Rank-Special Graphs Work Group". It is known that Z(G) ≥ δ(G), where δ(G) is the minimum degree of G. We show that Z(G) ≤ n - 3 if a connected graph G of order n has a connected complement graph [Formula: see text]. Further, we characterize a tree or a unicyclic graph G which satisfies either [Formula: see text] or [Formula: see text].

scholarly journals Zero forcing parameters and minimum rank problems

2010 ◽  
Vol 433 (2) ◽  
pp. 401-411 ◽  
Author(s):  
Francesco Barioli ◽  
Wayne Barrett ◽  
Shaun M. Fallat ◽  
H. Tracy Hall ◽  
Leslie Hogben ◽  
...  
Keyword(s):  
Zero Forcing ◽  
Minimum Rank

scholarly journals Maximum nullity and zero forcing of circulant graphs

2020 ◽  
Vol 8 (1) ◽  
pp. 221-234
Author(s):  
Linh Duong ◽  
Brenda K. Kroschel ◽  
Michael Riddell ◽  
Kevin N. Vander Meulen ◽  
Adam Van Tuyl
Keyword(s):  
Communication Complexity ◽  
Eigenvalue Problems ◽  
Electrical Power ◽  
Circulant Graphs ◽  
Zero Forcing ◽  
Hankel Matrices ◽  
Minimum Rank ◽  
Zero Forcing Number ◽  
Maximum Nullity ◽  
Forcing Number

AbstractThe zero forcing number of a graph has been applied to communication complexity, electrical power grid monitoring, and some inverse eigenvalue problems. It is well-known that the zero forcing number of a graph provides a lower bound on the minimum rank of a graph. In this paper we bound and characterize the zero forcing number of various circulant graphs, including families of bipartite circulants, as well as all cubic circulants. We extend the definition of the Möbius ladder to a type of torus product to obtain bounds on the minimum rank and the maximum nullity on these products. We obtain equality for torus products by employing orthogonal Hankel matrices. In fact, in every circulant graph for which we have determined these numbers, the maximum nullity equals the zero forcing number. It is an open question whether this holds for all circulant graphs.

scholarly journals Minimum rank and zero forcing number for butterfly networks

2018 ◽  
Vol 37 (3) ◽  
pp. 970-988 ◽  
Author(s):  
Daniela Ferrero ◽  
Cyriac Grigorious ◽  
Thomas Kalinowski ◽  
Joe Ryan ◽  
Sudeep Stephen
Keyword(s):  
Zero Forcing ◽  
Minimum Rank ◽  
Zero Forcing Number ◽  
Forcing Number
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