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.

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 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.

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 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 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).

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 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.

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