scholarly journals Signed zero forcing number and controllability for a networks system with a directed hypercube

2022 ◽  
Vol 355 ◽  
pp. 01012
Author(s):  
Gufang Mou ◽  
Qiuyan Zhang
Keyword(s):  
Effective Control ◽  
Signed Graph ◽  
Directed Network ◽  
Network System ◽  
Zero Forcing ◽  
Global Networks ◽  
Zero Forcing Number ◽  
Forcing Number ◽  
Minimum Number ◽  
The Relationship

The controllability for complex network system is to find the minimum number of leaders for the network system to achieve effective control of the global networks. In this paper, the problem of controllability of the directed network for a family of matrices carrying the structure under directed hypercube is considered. The relationship between the minimum number of leaders for the directed network system and the number of the signed zero forcing set is established. The minimum number of leaders of the directed networks system under a directed hypercube is obtained by computing the zero forcing number of a signed graph.

scholarly journals On the Relationships between Zero Forcing Numbers and Certain Graph Coverings

2014 ◽  
Vol 2 (1) ◽  
Author(s):  
Fatemeh Alinaghipour Taklimi ◽  
Shaun Fallat ◽  
Karen Meagher
Keyword(s):  
Upper Bound ◽  
Tree Cover ◽  
Positive Zero ◽  
Zero Forcing ◽  
Outerplanar Graphs ◽  
Zero Forcing Number ◽  
Forcing Number ◽  
Minimum Number ◽  
Graph Parameters ◽  
Graph Coverings

AbstractThe zero forcing number and the positive zero forcing number of a graph are two graph parameters that arise from two types of graph colourings. The zero forcing number is an upper bound on the minimum number of induced paths in the graph that cover all the vertices of the graph, while the positive zero forcing number is an upper bound on the minimum number of induced trees in the graph needed to cover all the vertices in the graph. We show that for a block-cycle graph the zero forcing number equals the path cover number.We also give a purely graph theoretical proof that the positive zero forcing number of any outerplanar graphs equals the tree cover number of the graph. These ideas are then extended to the setting of k-trees, where the relationship between the positive zero forcing number and the tree cover number becomes more complex.

scholarly journals Families of graphs with maximum nullity equal to zero forcing number

2018 ◽  
Vol 6 (1) ◽  
pp. 56-67
Author(s):  
Joseph S. Alameda ◽  
Emelie Curl ◽  
Armando Grez ◽  
Leslie Hogben ◽  
O’Neill Kingston ◽  
...  
Keyword(s):  
Color Change ◽  
Simple Graph ◽  
Zero Forcing ◽  
Generalized Petersen Graphs ◽  
Zero Forcing Number ◽  
Regularity Properties ◽  
Maximum Nullity ◽  
Forcing Number ◽  
Minimum Number ◽  
Petersen Graphs

Abstract The maximum nullity of a simple graph G, denoted M(G), is the largest possible nullity over all symmetric real matrices whose ijth entry is nonzero exactly when fi, jg is an edge in G for i =6 j, and the iith entry is any real number. The zero forcing number of a simple graph G, denoted Z(G), is the minimum number of blue vertices needed to force all vertices of the graph blue by applying the color change rule. This research is motivated by the longstanding question of characterizing graphs G for which M(G) = Z(G). The following conjecture was proposed at the 2017 AIM workshop Zero forcing and its applications: If G is a bipartite 3- semiregular graph, then M(G) = Z(G). A counterexample was found by J. C.-H. Lin but questions remained as to which bipartite 3-semiregular graphs have M(G) = Z(G). We use various tools to find bipartite families of graphs with regularity properties for which the maximum nullity is equal to the zero forcing number; most are bipartite 3-semiregular. In particular, we use the techniques of twinning and vertex sums to form new families of graphs for which M(G) = Z(G) and we additionally establish M(G) = Z(G) for certain Generalized Petersen 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.

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.

Zero forcing number of a graph in terms of the number of pendant vertices

2018 ◽  
Vol 68 (7) ◽  
pp. 1424-1433 ◽  
Author(s):  
Xinlei Wang ◽  
Dein Wong ◽  
Yuanshuai Zhang
Keyword(s):  
Zero Forcing ◽  
Zero Forcing Number ◽  
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 A lower bound on the zero forcing number

2018 ◽  
Vol 250 ◽  
pp. 363-367 ◽  
Author(s):  
Randy Davila ◽  
Thomas Kalinowski ◽  
Sudeep Stephen
Keyword(s):  
Lower Bound ◽  
Zero Forcing ◽  
Zero Forcing Number ◽  
Forcing Number
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