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

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

Discrete Applied Mathematics ◽

10.1016/j.dam.2020.03.027 ◽

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

Electronic Journal of Linear Algebra ◽

10.13001/1081-3810.2900 ◽

2015 ◽

Vol 30 ◽

Cited By ~ 6

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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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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