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.

Zero forcing number of fuzzy graphs with application

2020 ◽  
Vol 39 (3) ◽  
pp. 3873-3882
Author(s):  
Asefeh Karbasioun ◽  
R. Ameri
Keyword(s):  
Upper Bounds ◽  
Zero Forcing ◽  
Medical Treatments ◽  
Zero Forcing Number ◽  
Fuzzy Graphs ◽  
Forcing Number ◽  
Zero Forcing Numbers ◽  
Forcing Set ◽  
Zero Forcing Set

We introduce and study forcing number for fuzzy graphs. Also, we compute zero forcing numbers for some classes of graphs and extend this concept to fuzzy graphs. In this regard we obtain upper bounds for zero forcing of some classes of fuzzy graphs. We will proceed to obtain a new algorithm to computing zero forcing set and finding a formula for zero forcing number, and by some examples we illustrate these notions. Finally, we introduce some applications of fuzzy zero forcing in medical treatments.

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 Zero forcing number of degree splitting graphs and complete degree splitting graphs

2019 ◽  
Vol 11 (1) ◽  
pp. 40-53
Author(s):  
Charles Dominic
Keyword(s):  
Color Change ◽  
Simple Graph ◽  
White Vertex ◽  
Zero Forcing ◽  
Zero Forcing Number ◽  
Forcing Number ◽  
Minimum Number ◽  
Splitting Graph ◽  
Forcing Set ◽  
Zero Forcing Set

Abstract A subset ℤ ⊆ V(G) of initially colored black vertices of a graph G is known as a zero forcing set if we can alter the color of all vertices in G as black by iteratively applying the subsequent color change condition. At each step, any black colored vertex has exactly one white neighbor, then change the color of this white vertex as black. The zero forcing number ℤ (G), is the minimum number of vertices in a zero forcing set ℤ of G (see [11]). In this paper, we compute the zero forcing number of the degree splitting graph (𝒟𝒮-Graph) and the complete degree splitting graph (𝒞𝒟𝒮-Graph) of a graph. We prove that for any simple graph, ℤ [𝒟𝒮(G)] k + t, where ℤ (G) = k and t is the number of newly introduced vertices in 𝒟𝒮(G) to construct it.

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
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