scholarly journals Minimum Rank, Maximum Nullity and Zero Forcing Number of Graphs

2013 ◽  
pp. 775-810 ◽  
Author(s):  
Shaun Fallat
Keyword(s):  
Zero Forcing ◽  
Minimum Rank ◽  
Zero Forcing Number ◽  
Maximum Nullity ◽  
Forcing Number

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 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 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 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, maximum nullity, and zero forcing number of simple digraphs

2013 ◽  
Vol 26 ◽  
Author(s):  
Adam Berliner ◽  
Minerva Catral ◽  
Leslie Hogben ◽  
My Huynh ◽  
Kelsey Lied ◽  
...  
Keyword(s):  
Zero Forcing ◽  
Minimum Rank ◽  
Zero Forcing Number ◽  
Maximum Nullity ◽  
Forcing Number

scholarly journals Path cover number, maximum nullity, and zero forcing number of oriented graphs and other simple digraphs

2015 ◽  
Vol 8 (1) ◽  
pp. 147-167 ◽  
Author(s):  
Adam Berliner ◽  
Cora Brown ◽  
Joshua Carlson ◽  
Nathanael Cox ◽  
Leslie Hogben ◽  
...  
Keyword(s):  
Zero Forcing ◽  
Oriented Graphs ◽  
Path Cover ◽  
Zero Forcing Number ◽  
Maximum Nullity ◽  
Forcing Number ◽  
Path Cover Number

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

Maximum nullity and zero forcing number on graphs with maximum degree at most three

2020 ◽  
Vol 284 ◽  
pp. 179-194
Author(s):  
Meysam Alishahi ◽  
Elahe Rezaei-Sani ◽  
Elahe Sharifi
Keyword(s):  
Maximum Degree ◽  
Zero Forcing ◽  
Zero Forcing Number ◽  
Maximum Nullity ◽  
Forcing Number

scholarly journals Techniques for determining equality of the maximum nullity and the zero forcing number of a graph

2021 ◽  
Vol 37 ◽  
pp. 295-315
Author(s):  
Derek Young
Keyword(s):  
Lower Bound ◽  
Adjacency Matrix ◽  
Work Group ◽  
Zero Forcing ◽  
Vertex Connectivity ◽  
Aztec Diamond ◽  
Zero Forcing Number ◽  
Maximum Nullity ◽  
Forcing Number ◽  
Graph Parameters

It is known that the zero forcing number of a graph is an upper bound for the maximum nullity of the graph (see [AIM Minimum Rank - Special Graphs Work Group (F. Barioli, W. Barrett, S. Butler, S. Cioab$\breve{\text{a}}$, D. Cvetkovi$\acute{\text{c}}$, S. Fallat, C. Godsil, W. Haemers, L. Hogben, R. Mikkelson, S. Narayan, O. Pryporova, I. Sciriha, W. So, D. Stevanovi$\acute{\text{c}}$, H. van der Holst, K. Vander Meulen, and A. Wangsness). Linear Algebra Appl., 428(7):1628--1648, 2008]). In this paper, we search for characteristics of a graph that guarantee the maximum nullity of the graph and the zero forcing number of the graph are the same by studying a variety of graph parameters that give lower bounds on the maximum nullity of a graph. Inparticular, we introduce a new graph parameter which acts as a lower bound for the maximum nullity of the graph. As a result, we show that the Aztec Diamond graph's maximum nullity and zero forcing number are the same. Other graph parameters that are considered are a Colin de Verdiére type parameter and vertex connectivity. We also use matrices, such as a divisor matrix of a graph and an equitable partition of the adjacency matrix of a graph, to establish a lower bound for the nullity of the graph's adjacency matrix.

Using variants of zero forcing to bound the inertia set of a graph

2015 ◽  
Vol 30 ◽  
Author(s):  
Steve Butler ◽  
Jason Grout ◽  
H. Hall
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Minimal Cost ◽  
Zero Forcing ◽  
Combinatorial Game ◽  
Zero Forcing Number ◽  
Negative Eigenvalues ◽  
Maximum Nullity ◽  
Forcing Number

Zero forcing is a combinatorial game played on a graph with a goal of changing the color of every vertex at minimal cost. This leads to a parameter known as the zero forcing number that can be used to give an upper bound for the maximum nullity of a matrix associated with the graph. A variation on the zero forcing game is introduced that can be used to give an upper bound for the maximum nullity of such a matrix when it is constrained to have exactly q negative eigenvalues. This constrains the possible inertias that a matrix associated with a graph can achieve and gives a method to construct lower bounds on the inertia set of a graph (which is the set of all possible pairs (p,q) where p is the number of positive eigenvalues and q is the number of negative eigenvalues).

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