On zero forcing number of graphs and their complements

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].

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Zero Forcing Number of Some Families of Graphs

Mohyla Mathematical Journal ◽

10.18523/2617-70803202048-52cs ◽

2021 ◽

Vol 3 ◽

pp. 48-52

Author(s):

Victoria Petruk

Keyword(s):

Minimum Degree ◽

Colour Change ◽

Discrete Mathematics ◽

White Vertex ◽

Zero Forcing ◽

Minimum Cardinality ◽

Zero Forcing Number ◽

Ladder Graph ◽

Forcing Number ◽

Wheel Graph

The work is devoted to the study of the zero forcing number of some families of graphs. The concept of zero forcing is a relatively new research topic in discrete mathematics, which already has some practical applications, in particular, is used in studies of the minimum rank of the matrices of adjacent graphs. The zero forcing process is an example of the spreading process on graphs. Such processes are interesting not only in terms of mathematical and computer research, but also interesting and are used to model technical or social processes in other areas: statistical mechanics, physics, analysis of social networks, and so on. Let the vertices of the graph G be considered white, except for a certain set of S black vertices. We will repaint the vertices of the graph from white to black, using a certain rule.Colour change rule: A white vertex turns black if it is the only white vertex adjacent to the black vertex.[5] The zero forcing number Z(G) of the graph G is the minimum cardinality of the set of black vertices S required to convert all vertices of the graph G to black in a finite number of steps using the ”colour change rule”.It is known [10] that for any graph G, its zero forcing number cannot be less than the minimum degree of its vertices. Such and other already known facts became the basis for finding the zero forcing number for two given below families of graphs:A gear graph, denoted W2,n is a graph obtained by inserting an extra vertex between each pair of adjacent vertices on the perimeter of a wheel graph Wn. Thus, W2,n has 2n + 1 vertices and 3n edges.A prism graph, denoted Yn, or in general case Ym,n, and sometimes also called a circular ladder graph, is a graph corresponding to the skeleton of an n-prism.A wheel graph, denoted Wn is a graph formed by connecting a single universal vertex to all vertices of a cycle of length n.In this article some known results are reviewed, there is also a definition, proof and some examples of the zero forcing number and the zero forcing process of gear graphs and prism graphs.

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

Acta Universitatis Sapientiae Mathematica ◽

10.2478/ausm-2019-0004 ◽

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.

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The Bipartite Zero Forcing Set for a Full Sign Pattern Matrix

Mathematics ◽

10.3390/math8030354 ◽

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.

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Critical ideals, minimum rank and zero forcing number

Applied Mathematics and Computation ◽

10.1016/j.amc.2019.04.043 ◽

2019 ◽

Vol 358 ◽

pp. 305-313 ◽

Cited By ~ 1

Author(s):

Carlos A. Alfaro ◽

Jephian C.-H. Lin

Keyword(s):

Zero Forcing ◽

Minimum Rank ◽

Zero Forcing Number ◽

Forcing Number

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

Linear Algebra and its Applications ◽

10.1016/j.laa.2010.10.015 ◽

2012 ◽

Vol 436 (12) ◽

pp. 4352-4372 ◽

Cited By ~ 39

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

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

Electronic Journal of Linear Algebra ◽

10.13001/ela.2021.4967 ◽

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.

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

Involve a Journal of Mathematics ◽

10.2140/involve.2010.3.371 ◽

2010 ◽

Vol 3 (4) ◽

pp. 371-392 ◽

Cited By ~ 3

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

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Families of graphs with maximum nullity equal to zero forcing number

Special Matrices ◽

10.1515/spma-2018-0006 ◽

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.

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Maximum nullity and zero forcing of circulant graphs

Special Matrices ◽

10.1515/spma-2020-0106 ◽

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.

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