Minimum Rank, Maximum Nullity, and Zero Forcing Number for
Small Digraphs
∗
Josh Carlson
Iowa State University, Ames, IA
Introduction
Useful Techniques
Field Independence
Minimum rank, maximum nullity, and zero forcing number have been thoroughly studied for
simple graphs. We calculate the values of these parameters for all simple digraphs on 4
vertices and less. We illustrate some useful techniques to find these values and examine
the concept of field independence of minimum rank. We also exhibit a digraph whose
minimum rank and maximum nullity have a unique property.
There are three main techniques that are very helpful in calculating the values of the zero forcing number and minimum
rank (thus maximum nullity) of all 238 digraphs of order 4 or less:
If not specified, mr(Γ ) is taken to be the minimum rank of a digraph Γ over R. The minimum
rank of Hessenberg paths and ditrees are field independent. This means we only need to
consider field independence for the digraphs that are not Hessenberg paths or ditrees.
Definitions
I
I
I
Hessenberg paths
I Ditrees
I Finding a realizing matrix
I
Zero Forcing Number
A graph G is defined as G = (V , E) where V is a set of vertices and E is set of ordered
pairs of two distinct elements of V ; (i, j) ∈ E represents the edge connecting vertex i to
vertex j.
For small digraphs we can quickly determine the zero forcing number by making sure there is no zero forcing set of size
k − 1 and exhibiting a zero forcing set of size k .
Given n × n matrix A , Γ (A ) is the digraph of n vertices that is associated with A where
there is an arc from verticex i to vertex j if and only if the ij entry of A is non-zero. The
diagonal of A is free.

0
 −1

 1
0
1
2
−1
3
0
0
0
2
0
1

0
1
0
?
0
1
1
1
?
1
1
0

0
?
Matrix D
Matrix C
4
3
This is an example of a digraph that has field independent minimum rank:


0 0 0 ∗
1
∗ 0 ∗ 0


∗ 0 ∗ ∗
2
4
0 0 0 0
Hessenberg Paths
There are two theorems about Hessenberg paths that were beneficial to us:
Γ (A )
‘
I
Given a digraph Γ , M(Γ ) is the set of all n × n matrices such that Γ (A ) = Γ .
I
The minimum rank of a digraph Γ is defined as mr(Γ ) = min{rank (A ) : A ∈ M(Γ )}.
I
The maximum nullity of a digraph Γ is defined as M(Γ ) = max{null(A ) : A ∈ M(Γ )}.
Theorem[1]: Z(Γ ) = 1 if and only if Γ is a Hessenberg path.
Theorem[1]: If Γ is a Hessenberg path of n vertices then mr(Γ ) = n − 1.
3
Matrix E
(D87)[3]
1
The stars in matrix E are non-zero entries. Matrix E corresponds to D87 because
Γ (E) = D87. Rows 1 and 2 of Matrix E are linearly independent. In any field we can make
row 3 equal to the sum of rows 1 and 2. This means that mrF (D87) = 2 for any field F which
makes it field independent.
Observation: For every digraph Γ with n vertices, mr(Γ ) + M(Γ ) = n.
I
?
1

0
0

There is no zero forcing set of size 1 but vertex 1 and 4 are a zero forcing set so Z(D163) = 2.
2
Matrix A
3
(D163)[3]
1


Since Z(D122) = 2, mz(D122) = 2. Matrices C and D both correspond to D122. In R
matrix C has rank 2 because rows 1 and 2 are linearly independent and rows 3 and 4
depend on them. This means that mr R(D122) = 2 by the realizing matrix argument. If we
consider matrix D in Z2, the off-diagonal entries are fixed because in Z2 there is only one
non-zero entry. Rows 1, 2, and 4 are linearly independent given all possible diagonal
entries. This means that in Z2 matrix D has to have rank 3. Thus mr Z2 (D122) = 3 and
D122 has field dependent minimum rank.
2
4
4
3
(D122)[3]
1
A digraph Γ is defined as Γ = (V, E) where V is a set of vertices and E is set of ordered
pairs of two distinct elements of V ; (i, j) ∈ E represents the arc pointing from vertex i to
vertex j.
This is an example of a digraph that has field dependent minimum rank:


1
1 0 1 1
 −1 1 1 0 


 0 0 0 0
2
0 1 2 1
2
The color change rule of a digraph Γ states that if a vertex b is blue and a white vertex w is
the only white out-neighbor of b, then b can force w to be colored blue.
3
4
Only 13 of the 238 digraphs of order 4 or less have field dependent minimum rank.
(D201)[3]
b
w
b
w
‘
Vertex 3 is a zero forcing set so Z(D201) = 1. By the two theorems, D201 is a Hessenberg path so mr(D201) = 1 and
M(D201) = 3.
A Particularly Interesting Digraph
1
b can force w
b cannot force w
Trees
A zero forcing set of a digraph Γ is a set of vertices such that when initially colored blue, the
entire digraph can be forced according to the zero forcing rule.
We have one very useful tool for determining the minimum rank of ditrees:
I The zero forcing number Z(Γ ) is the minimum number of blue vertices initially needed to
Theorem[2]: If T is a ditree then Z(T ) = M(T ).
force all vertices in Γ blue according to the color change rule.
I
2
A Hessenberg path Γ is path where E(Γ ) does not contain any arc of the form (vi , vj ) with
j >i+1
3
Z(D162) = 2 and mr(D162) = 3 (which means M(D162) = n − 3 = 1). We know that
M(Γ ) 6 Z(Γ ) for any digraph Γ . D162 happened to be the only digraph of order 4 or less that
had M(Γ ) , Z(Γ ). D162 also happens to be one of the 13 digraphs with field dependent
minimum rank.
4
(D70)[3]
Further Questions
Since D70 is a ditree, M(D70) = Z(D70) = 2 and mr(D70) = n − 2 = 2.
The red arc is not allowed
I
I
I
An undirected cycle is a sequence of vertices v1, v2, . . . , vn such that vi−1vi is an edge for
i = 2, . . . , n and (vn , v1) is an edge.
An undirected tree is a connected graph that contains no cycles.
A directed tree (ditree) is a digraph whose underlying graph is a tree.
Finding a Realizing Matrix
If k is a lower bound for mr(Γ ) and we can find a matrix that corresponds to Γ with rank k , then we know mr(Γ ) = k . Since
mz(Γ ) 6 mr(Γ ), all we have to do is find such a matrix with k = mz(Γ ).


0 1 1 0
1
0 1 1 0


 1 −1 0 0 
4
2
1 0 1 0
3
(D181)[3]
Cycle
I
Tree
Ditree
The miminum rank of a digraph Γ is field independent if mrF (Γ ) = mr R(Γ ) for any field F.
4
(D162)[3]
For a digraph Γ , the parameter mz(Γ ) is defined as n − Z(Γ ).
Observation: mz(Γ ) 6 mr(Γ ) for all digraphs Γ
I
3
1
Observation: M(Γ ) 6 Z(Γ ) for all digraphs Γ .
I
2
Matrix B
Since Z(D181) = 2, mz(D181) = n − 2 = 2. Matrix B corresponds to D181 because Γ (B) = D181. Rank (B) = 2 because
rows 2 and 3 are linearly independent and rows 1 and 4 are dependent on them. Since rank (B) = mz(D181) = 2,
mr(D181) = 2.
Joint work with: Adam Berliner, Cora Brown, Nathanael Cox, Leslie Hogben, Jason Hu, Katrina Jacobs, Kathryn Manternach, Travis Peters, Nathan Warnberg, and Michael Young
I
What is the minimum rank, maximum nullity, and zero forcing number for the digraphs of
order 5?
I
How many digraphs of order 5 have field dependent minimum rank?
I
If Γ is a digraph and M(Γ ) = 1, what are the possible values of Z(Γ )?
References
[1] A. Berliner, M. Catral, L. Hogben, M. Huynh, K. Lied, M. Young. Minimum rank, maximum
nullity, and zero forcing number for simple digraphs. Under review.
[2] L. Hogben. Minimum rank problems. Linear Algebra and its Applications, 432:
1961-1974, 2010.
[3] R. C. Read and R. J. Wilson. An Atlas of Graphs, Oxford University Press, New York,
1998.
∗
Department of Mathematics, Iowa State University, Ames, IA 50011 (jmsdg7@iastate.edu)