Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Lights Out on Related Graphs
Laura Ballard
Hope College REU withErica Budge and Dr. Darin Stephenson
Summer 2012
October 16, 2015
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
What is “Lights Out”?
The original Tiger Electronics game:
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
What is “Lights Out”?
The original Tiger Electronics game:
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
What is “Lights Out”?
Each light is a vertex
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
What is “Lights Out”?
Connections are edges
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
What is “Lights Out”?
In the Z2 case, 1 is “on” and 0 is “o↵”
Blankverticesare instate 0.
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
What is “Lights Out”?
In the Z2 case, 1 is “on” and 0 is “o↵”
Blankverticesare instate 0.
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
What is “Lights Out”?
The finished product:
1 1
1 1 1
1 1
1
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
What is “Lights Out”?
Another graph:
1
1 1
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
What is “Lights Out”?
Definition
A press on a vertex v increases the state of v and all adjacentvertices by 1.
In Z2,1 + 1 = 0.
1
1 1
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
What is “Lights Out”?
Definition
A press on a vertex v increases the state of v and all adjacentvertices by 1.
In Z2,1 + 1 = 0.
1
1 1
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
What is “Lights Out”?
Definition
A press on a vertex v increases the state of v and all adjacentvertices by 1.
In Z2,1 + 1 = 0.
1 1
1 1
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
What is “Lights Out”?
Definition
A press on a vertex v increases the state of v and all adjacentvertices by 1.
In Z2,1 + 1 = 0.
1 1
1 1
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
What is “Lights Out”?
Definition
A press on a vertex v increases the state of v and all adjacentvertices by 1.
In Z2,1 + 1 = 0.
1
1 1
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
What is “Lights Out”?
Remember, the objective is to turn all of the lights out.
1
1 1
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
What is “Lights Out”?
Lights out!!
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
It gets better!
But we don’t just work in Z2...
In Zk , possible states are between 0 and k � 1.
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
It gets better!
But we don’t just work in Z2...In Zk , possible states are between 0 and k � 1.
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
In Z4:
But we don’t just work in Z2...In Zk , possible states are between 0 and k � 1.
3
2
3 1
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
In Z4:
3
2
3 1
In Z4,3 + 1 = 0.
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
In Z4:
3
2
3 1
In Z4,3 + 1 = 0.
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
In Z4:
1
3
3 1
In Z4,3 + 1 = 0.
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
For multiple presses:
A pattern is a vector that represents multiple presses on agraph G .
For example, pattern ~y would press A twice and everyother vertex once:
A
B C D
E
~y =
0
BBBB@
21111
1
CCCCA
ABCDE
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
For multiple presses:
A pattern is a vector that represents multiple presses on agraph G .
For example, pattern ~y would press A twice and everyother vertex once:
A
B C D
E
~y =
0
BBBB@
21111
1
CCCCA
ABCDE
L. Ballard (Syracuse University) Lights Out on Related Graphs October 16, 2015 19 / 53
Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
For multiple presses:
A pattern is a vector that represents multiple presses on agraph G .
For example, pattern ~y would press A twice and everyother vertex once:
A
B C D
E
~y =
0
BBBB@
21111
1
CCCCA
ABCDE
L. Ballard (Syracuse University) Lights Out on Related Graphs October 16, 2015 19 / 53
Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
For multiple presses:
A pattern is a vector that represents multiple presses on agraph G .
For example, pattern ~y would press A twice and everyother vertex once:
A
B C D
E
~y =
0
BBBB@
21111
1
CCCCA
ABCDE
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Definition
If a state can be turned all o↵ by applying a pattern, that stateis a winnable state.
B C
A
1
D
1
E F
1
0
BBBBBB@
100101
1
CCCCCCA
ABCDEF
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Definition
If a state can be turned all o↵ by applying a pattern, that stateis a winnable state.
B C
A
1
D
1
E F
1
0
BBBBBB@
100101
1
CCCCCCA
ABCDEF
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Definition
If a state can be turned all o↵ by applying a pattern, that stateis a winnable state.
B C
A1
D1
E F1
0
BBBBBB@
100101
1
CCCCCCA
ABCDEF
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Definition
If a state can be turned all o↵ by applying a pattern, that stateis a winnable state.
B C
A1
D1
E F1
0
BBBBBB@
100101
1
CCCCCCA
ABCDEF
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Definition
If a state can be turned all o↵ by applying a pattern, that stateis a winnable state.
B C
A1
D1
E F1
0
BBBBBB@
100101
1
CCCCCCA
ABCDEF
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Linear Algebra
A
B C D
E
G
Neighborhood Matrix of G(denoted N(G ))
A B C D EA 1 1 1 1 0B 1 1 1 0 1C 1 1 1 1 1D 1 0 1 1 1E 0 1 1 1 1
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Linear Algebra
A
B C D
E
G
Neighborhood Matrix of G(denoted N(G ))
A B C D EA 1 1 1 1 0B 1 1 1 0 1C 1 1 1 1 1D 1 0 1 1 1E 0 1 1 1 1
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Linear Algebra
A
B C D
E
G
Neighborhood Matrix of G(denoted N(G ))
A B C D EA
1 1 1 1 0
B
1 1 1 0 1
C
1 1 1 1 1
D
1 0 1 1 1
E
0 1 1 1 1
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Linear Algebra
A
B C D
E
G
Neighborhood Matrix of G(denoted N(G ))
A B C D EA
1
1 1 1
0
B 1
1
1
0
1C 1 1
1
1 1D 1
0
1
1
1E
0
1 1 1
1
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Linear Algebra
A
B C D
E
G
Neighborhood Matrix of G(denoted N(G ))
A B C D EA
1
1 1 1 0B 1
1
1 0 1C 1 1
1
1 1D 1 0 1
1
1E 0 1 1 1
1
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Linear Algebra
A
B C D
E
G
Neighborhood Matrix of G(denoted N(G ))
A B C D EA 1 1 1 1 0B 1 1 1 0 1C 1 1 1 1 1D 1 0 1 1 1E 0 1 1 1 1
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Linear Algebra
A
B C D
E
G
Neighborhood Matrix of G(denoted N(G ))
0
BBBB@
1 1 1 1 01 1 1 0 11 1 1 1 11 0 1 1 10 1 1 1 1
1
CCCCA
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Linear Algebra
The “initial state” of a graph can also be represented as avector ~x .
2
A
0
B
1
C
2
D
2
E
In Z3:~x =
0
BBBB@
20122
1
CCCCA
ABCDE
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Linear Algebra
The “initial state” of a graph can also be represented as avector ~x .
2A
0B
1C
2D
2E
In Z3:
~x =
0
BBBB@
20122
1
CCCCA
ABCDE
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Linear Algebra
The “initial state” of a graph can also be represented as avector ~x .
2A
0B
1C
2D
2E
In Z3:~x =
0
BBBB@
20122
1
CCCCA
ABCDE
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Applying Patterns
N~y gives the e↵ect of applying pattern ~y to graph G .
N~y =
0
BBBB@
1 1 1 1 01 1 1 0 11 1 1 1 11 0 1 1 10 1 1 1 1
1
CCCCA
0
BBBB@
21111
1
CCCCA=
2
0
BBBB@
11110
1
CCCCA+ 1
0
BBBB@
11101
1
CCCCA+ 1
0
BBBB@
11111
1
CCCCA+ 1
0
BBBB@
10111
1
CCCCA+ 1
0
BBBB@
01111
1
CCCCA
=
0
BBBB@
21211
1
CCCCA
}Pressing A
Pressing vertices in a pattern is commutative.
For every pattern ~y , N~y 2 CSk(G )
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Applying Patterns
N~y gives the e↵ect of applying pattern ~y to graph G .
N~y =
0
BBBB@
1 1 1 1 01 1 1 0 11 1 1 1 11 0 1 1 10 1 1 1 1
1
CCCCA
0
BBBB@
21111
1
CCCCA=
2
0
BBBB@
11110
1
CCCCA+ 1
0
BBBB@
11101
1
CCCCA+ 1
0
BBBB@
11111
1
CCCCA+ 1
0
BBBB@
10111
1
CCCCA+ 1
0
BBBB@
01111
1
CCCCA
=
0
BBBB@
21211
1
CCCCA
}Pressing A
Pressing vertices in a pattern is commutative.
For every pattern ~y , N~y 2 CSk(G )
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Applying Patterns
N~y gives the e↵ect of applying pattern ~y to graph G .
N~y =
0
BBBB@
1 1 1 1 01 1 1 0 11 1 1 1 11 0 1 1 10 1 1 1 1
1
CCCCA
0
BBBB@
21111
1
CCCCA=
2
0
BBBB@
11110
1
CCCCA+ 1
0
BBBB@
11101
1
CCCCA+ 1
0
BBBB@
11111
1
CCCCA+ 1
0
BBBB@
10111
1
CCCCA+ 1
0
BBBB@
01111
1
CCCCA
=
0
BBBB@
21211
1
CCCCA}Pressing A
Pressing vertices in a pattern is commutative.
For every pattern ~y , N~y 2 CSk(G )
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Applying Patterns
N~y gives the e↵ect of applying pattern ~y to graph G .
N~y =
0
BBBB@
1 1 1 1 01 1 1 0 11 1 1 1 11 0 1 1 10 1 1 1 1
1
CCCCA
0
BBBB@
21111
1
CCCCA=
2
0
BBBB@
11110
1
CCCCA+ 1
0
BBBB@
11101
1
CCCCA+ 1
0
BBBB@
11111
1
CCCCA+ 1
0
BBBB@
10111
1
CCCCA+ 1
0
BBBB@
01111
1
CCCCA
=
0
BBBB@
21211
1
CCCCA
}Pressing A
Pressing vertices in a pattern is commutative.
For every pattern ~y , N~y 2 CSk(G )
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Applying Patterns
N~y gives the e↵ect of applying pattern ~y to graph G .
N~y =
0
BBBB@
1 1 1 1 01 1 1 0 11 1 1 1 11 0 1 1 10 1 1 1 1
1
CCCCA
0
BBBB@
21111
1
CCCCA=
2
0
BBBB@
11110
1
CCCCA+ 1
0
BBBB@
11101
1
CCCCA+ 1
0
BBBB@
11111
1
CCCCA+ 1
0
BBBB@
10111
1
CCCCA+ 1
0
BBBB@
01111
1
CCCCA
=
0
BBBB@
21211
1
CCCCA
}Pressing A
Pressing vertices in a pattern is commutative.
For every pattern ~y , N~y 2 CSk(G )
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Applying Patterns
N~y gives the e↵ect of applying pattern ~y to graph G .
N~y =
0
BBBB@
1 1 1 1 01 1 1 0 11 1 1 1 11 0 1 1 10 1 1 1 1
1
CCCCA
0
BBBB@
21111
1
CCCCA=
2
0
BBBB@
11110
1
CCCCA+ 1
0
BBBB@
11101
1
CCCCA+ 1
0
BBBB@
11111
1
CCCCA+ 1
0
BBBB@
10111
1
CCCCA+ 1
0
BBBB@
01111
1
CCCCA
=
0
BBBB@
21211
1
CCCCA
}Pressing A
Pressing vertices in a pattern is commutative.
For every pattern ~y , N~y 2 CSk(G )
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Applying Patterns
N~y gives the e↵ect of applying pattern ~y to graph G .
N~y =
0
BBBB@
1 1 1 1 01 1 1 0 11 1 1 1 11 0 1 1 10 1 1 1 1
1
CCCCA
0
BBBB@
21111
1
CCCCA=
2
0
BBBB@
11110
1
CCCCA+ 1
0
BBBB@
11101
1
CCCCA+ 1
0
BBBB@
11111
1
CCCCA+ 1
0
BBBB@
10111
1
CCCCA+ 1
0
BBBB@
01111
1
CCCCA=
0
BBBB@
21211
1
CCCCA}Pressing A
Pressing vertices in a pattern is commutative.
For every pattern ~y , N~y 2 CSk(G )
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Applying Patterns
The new state of G after pattern ~y is complete is N~y + ~x .
0
BBBB@
21211
1
CCCCA+
0
BBBB@
20122
1
CCCCA=
0
BBBB@
41333
1
CCCCAmod 3 =
0
BBBB@
11000
1
CCCCA
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Applying Patterns
The new state of G after pattern ~y is complete is N~y + ~x .
0
BBBB@
21211
1
CCCCA+
0
BBBB@
20122
1
CCCCA=
0
BBBB@
41333
1
CCCCAmod 3 =
0
BBBB@
11000
1
CCCCA
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Applying Patterns
The new state of G after pattern ~y is complete is N~y + ~x .
0
BBBB@
21211
1
CCCCA+
0
BBBB@
20122
1
CCCCA=
0
BBBB@
41333
1
CCCCAmod 3
=
0
BBBB@
11000
1
CCCCA
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Applying Patterns
The new state of G after pattern ~y is complete is N~y + ~x .
0
BBBB@
21211
1
CCCCA+
0
BBBB@
20122
1
CCCCA=
0
BBBB@
41333
1
CCCCAmod 3 =
0
BBBB@
11000
1
CCCCA
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
When we apply patterns, the goal is to change each vertex tostate 0:
N~y + ~x = ~0
N~y = �~x
~y = N�1 ⇤ (�~x)
If N is invertible mod k , every initial state ~x is winnable, andN�1 ⇤ (�~x) is the winning pattern.
Graphs for which N is invertible mod k are called always
winnable mod k .
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
When we apply patterns, the goal is to change each vertex tostate 0:
N~y + ~x = ~0
N~y = �~x
~y = N�1 ⇤ (�~x)
If N is invertible mod k , every initial state ~x is winnable, andN�1 ⇤ (�~x) is the winning pattern.
Graphs for which N is invertible mod k are called always
winnable mod k .
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
When we apply patterns, the goal is to change each vertex tostate 0:
N~y + ~x = ~0
N~y = �~x
~y = N�1 ⇤ (�~x)
If N is invertible mod k , every initial state ~x is winnable, andN�1 ⇤ (�~x) is the winning pattern.
Graphs for which N is invertible mod k are called always
winnable mod k .
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ConclusionAcknowledg-mentsReferences
When we apply patterns, the goal is to change each vertex tostate 0:
N~y + ~x = ~0
N~y = �~x
~y = N�1 ⇤ (�~x)
If N is invertible mod k , every initial state ~x is winnable, andN�1 ⇤ (�~x) is the winning pattern.
Graphs for which N is invertible mod k are called always
winnable mod k .
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L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
When we apply patterns, the goal is to change each vertex tostate 0:
N~y + ~x = ~0
N~y = �~x
~y = N�1 ⇤ (�~x)
If N is invertible mod k , every initial state ~x is winnable, andN�1 ⇤ (�~x) is the winning pattern.
Graphs for which N is invertible mod k are called always
winnable mod k .
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L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
When we apply patterns, the goal is to change each vertex tostate 0:
N~y + ~x = ~0
N~y = �~x
~y = N�1 ⇤ (�~x)
If N is invertible mod k , every initial state ~x is winnable, andN�1 ⇤ (�~x) is the winning pattern.
Graphs for which N is invertible mod k are called always
winnable mod k .
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Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Remember that N~y 2 CSk(G ).
N~y = �~x is solvable if ~x is winnable.
Thus, ~x is winnable if and only if ~x 2 CSk(G ).
(Null Space of G - NSk(G ))
Since N is a symmetric matrix, NSk(G ) is the orthogonalcomplement of CSk(G ).
A graph is always winnable mod k if and only ifdimNSk(G ) = 0.
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Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Remember that N~y 2 CSk(G ).
N~y = �~x is solvable if ~x is winnable.
Thus, ~x is winnable if and only if ~x 2 CSk(G ).
(Null Space of G - NSk(G ))
Since N is a symmetric matrix, NSk(G ) is the orthogonalcomplement of CSk(G ).
A graph is always winnable mod k if and only ifdimNSk(G ) = 0.
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Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Remember that N~y 2 CSk(G ).
N~y = �~x is solvable if ~x is winnable.
Thus, ~x is winnable if and only if ~x 2 CSk(G ).
(Null Space of G - NSk(G ))
Since N is a symmetric matrix, NSk(G ) is the orthogonalcomplement of CSk(G ).
A graph is always winnable mod k if and only ifdimNSk(G ) = 0.
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L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Remember that N~y 2 CSk(G ).
N~y = �~x is solvable if ~x is winnable.
Thus, ~x is winnable if and only if ~x 2 CSk(G ).
(Null Space of G - NSk(G ))
Since N is a symmetric matrix, NSk(G ) is the orthogonalcomplement of CSk(G ).
A graph is always winnable mod k if and only ifdimNSk(G ) = 0.
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L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Remember that N~y 2 CSk(G ).
N~y = �~x is solvable if ~x is winnable.
Thus, ~x is winnable if and only if ~x 2 CSk(G ).
(Null Space of G - NSk(G ))
Since N is a symmetric matrix, NSk(G ) is the orthogonalcomplement of CSk(G ).
A graph is always winnable mod k if and only ifdimNSk(G ) = 0.
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L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Our Research Focus
~x is winnable if and only if ~x 2 CSk(G ).
A graph is always winnable mod k if and only ifdimNSk(G ) = 0.
NSk(G ) is the orthogonal complement of CSk(G ).
Characterize the way the null space changes when certainsubgraphs are removed, or when two graphs are joinedtogether.
Look for ways to connect graphs or remove subgraphs thatdo not change the null space.
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Our Research Focus
~x is winnable if and only if ~x 2 CSk(G ).
A graph is always winnable mod k if and only ifdimNSk(G ) = 0.
NSk(G ) is the orthogonal complement of CSk(G ).
Characterize the way the null space changes when certainsubgraphs are removed, or when two graphs are joinedtogether.
Look for ways to connect graphs or remove subgraphs thatdo not change the null space.
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Our Research Focus
~x is winnable if and only if ~x 2 CSk(G ).
A graph is always winnable mod k if and only ifdimNSk(G ) = 0.
NSk(G ) is the orthogonal complement of CSk(G ).
Characterize the way the null space changes when certainsubgraphs are removed, or when two graphs are joinedtogether.
Look for ways to connect graphs or remove subgraphs thatdo not change the null space.
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Our Research Focus
~x is winnable if and only if ~x 2 CSk(G ).
A graph is always winnable mod k if and only ifdimNSk(G ) = 0.
NSk(G ) is the orthogonal complement of CSk(G ).
Characterize the way the null space changes when certainsubgraphs are removed, or when two graphs are joinedtogether.
Look for ways to connect graphs or remove subgraphs thatdo not change the null space.
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Our Research Focus
~x is winnable if and only if ~x 2 CSk(G ).
A graph is always winnable mod k if and only ifdimNSk(G ) = 0.
NSk(G ) is the orthogonal complement of CSk(G ).
Characterize the way the null space changes when certainsubgraphs are removed, or when two graphs are joinedtogether.
Look for ways to connect graphs or remove subgraphs thatdo not change the null space.
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
The Null Space
By definition, if ~z 2 NSk(G ), N~z = ~0
Then ~z is a null pattern that will not change the state ofgraph G , since N~z + ~x = ~x .
B
C
A
D
~z =
0
BB@
1100
1
CCA
ABCD
In Z2
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
The Null Space
By definition, if ~z 2 NSk(G ), N~z = ~0
Then ~z is a null pattern that will not change the state ofgraph G , since N~z + ~x = ~x .
B
C
A
D
~z =
0
BB@
1100
1
CCA
ABCD
In Z2
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
The Null Space
By definition, if ~z 2 NSk(G ), N~z = ~0
Then ~z is a null pattern that will not change the state ofgraph G , since N~z + ~x = ~x .
B
C
A
D
~z =
0
BB@
1100
1
CCA
ABCD
In Z2
L. Ballard (Syracuse University) Lights Out on Related Graphs October 16, 2015 29 / 53
Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
The Null Space
By definition, if ~z 2 NSk(G ), N~z = ~0
Then ~z is a null pattern that will not change the state ofgraph G , since N~z + ~x = ~x .
B
C
A
D
~z =
0
BB@
1100
1
CCA
ABCD
In Z2
L. Ballard (Syracuse University) Lights Out on Related Graphs October 16, 2015 29 / 53
Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
The Null Space
By definition, if ~z 2 NSk(G ), N~z = ~0
Then ~z is a null pattern that will not change the state ofgraph G , since N~z + ~x = ~x .
B
C
A
D
~z =
0
BB@
1100
1
CCA
ABCD
In Z2
L. Ballard (Syracuse University) Lights Out on Related Graphs October 16, 2015 29 / 53
Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
The Null Space
By definition, if ~z 2 NSk(G ), N~z = ~0
Then ~z is a null pattern that will not change the state ofgraph G , since N~z + ~x = ~x .
B
C
A
D
~z =
0
BB@
1100
1
CCA
ABCD
In Z2
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
The Null Space
By definition, if ~z 2 NSk(G ), N~z = ~0
Then ~z is a null pattern that will not change the state ofgraph G , since N~z + ~x = ~x .
1
1
1
B
C
A
D
~z =
0
BB@
1100
1
CCA
ABCD
In Z2
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
The Null Space
By definition, if ~z 2 NSk(G ), N~z = ~0
Then ~z is a null pattern that will not change the state ofgraph G , since N~z + ~x = ~x .
1
1
1
B
C
A
D
~z =
0
BB@
1100
1
CCA
ABCD
In Z2
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
The Null Space
By definition, if ~z 2 NSk(G ), N~z = ~0
Then ~z is a null pattern that will not change the state ofgraph G , since N~z + ~x = ~x .
B
C
A
D
~z =
0
BB@
1100
1
CCA
ABCD
In Z2
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L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
The Null Space
This null pattern can be generalized to Zk .
B
C
A
D
~z =
0
BB@
t�t00
1
CCA
ABCD
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
The Null Space
This null pattern can be generalized to Zk .
B
C
A
D
~z =
0
BB@
t�t00
1
CCA
ABCD
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
The Null Space
This null pattern can be generalized to Zk .
t
t
t
B
C
A
D
~z =
0
BB@
t�t00
1
CCA
ABCD
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
The Null Space
This null pattern can be generalized to Zk .
t
t
t
B
C
A
D
~z =
0
BB@
t�t00
1
CCA
ABCD
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
The Null Space
This null pattern can be generalized to Zk .
B
C
A
D
~z =
0
BB@
t�t00
1
CCA
ABCD
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L. Ballard
Lights OutIntroductionLinear Algebra
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ConclusionAcknowledg-mentsReferences
A Sampling of Our Results
Definition
The label of a vertex v in a graph G will be defined by
lG (v) = dimNSk(G � v)� dimNSk(G ).
In other words, dimNSk(G � v) = dimNSk(G ) + lG (v)
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A Sampling of Our Results
Proposition
Let G be a graph. For all v 2 V (G ), we havelG (v) 2 {�1, 0, 1}. If G is always winnable over Zk , then forall v 2 V (G ), we have lG (v) 2 {0, 1}.
Proof. The neighborhood matrix of G � v is formed by deletingexactly one row and one column from the neighborhood matrixof G . Thus lG (v) 2 {�1, 0, 1}. For an always winnable graphG , dimNSk(G ) = 0, and therefore for all v 2 V (G ), we havedimNSk(G � v) � dimNSk(G ), so lG (v) 2 {0, 1}.
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Lights OutIntroductionLinear Algebra
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A Sampling of Our Results
Proposition
Let G be a graph. For all v 2 V (G ), we havelG (v) 2 {�1, 0, 1}. If G is always winnable over Zk , then forall v 2 V (G ), we have lG (v) 2 {0, 1}.
Proof. The neighborhood matrix of G � v is formed by deletingexactly one row and one column from the neighborhood matrixof G .
Thus lG (v) 2 {�1, 0, 1}. For an always winnable graphG , dimNSk(G ) = 0, and therefore for all v 2 V (G ), we havedimNSk(G � v) � dimNSk(G ), so lG (v) 2 {0, 1}.
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Lights OutIntroductionLinear Algebra
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ConclusionAcknowledg-mentsReferences
A Sampling of Our Results
Proposition
Let G be a graph. For all v 2 V (G ), we havelG (v) 2 {�1, 0, 1}. If G is always winnable over Zk , then forall v 2 V (G ), we have lG (v) 2 {0, 1}.
Proof. The neighborhood matrix of G � v is formed by deletingexactly one row and one column from the neighborhood matrixof G . Thus lG (v) 2 {�1, 0, 1}.
For an always winnable graphG , dimNSk(G ) = 0, and therefore for all v 2 V (G ), we havedimNSk(G � v) � dimNSk(G ), so lG (v) 2 {0, 1}.
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ConclusionAcknowledg-mentsReferences
A Sampling of Our Results
Proposition
Let G be a graph. For all v 2 V (G ), we havelG (v) 2 {�1, 0, 1}. If G is always winnable over Zk , then forall v 2 V (G ), we have lG (v) 2 {0, 1}.
Proof. The neighborhood matrix of G � v is formed by deletingexactly one row and one column from the neighborhood matrixof G . Thus lG (v) 2 {�1, 0, 1}. For an always winnable graphG , dimNSk(G ) = 0, and therefore for all v 2 V (G ), we havedimNSk(G � v) � dimNSk(G ), so lG (v) 2 {0, 1}.
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A Sampling of Our Results
Proposition
If there exists ~p 2 NSk(G ) with ~p(v) 6= 0, then lG (v) = �1.
Suppose ~p 2 NSk(G ) such that ~p(v) 6= 0. Then ev is notwinnable on G , since ev is not in the orthogonal complementof NSk(G ). It follows that a pattern on G that is null on G � vis also null at v ; otherwise that pattern would win �ev for some� 2 Z⇤
k . In particular, a null pattern on G � v extended by 0 atv is null on G , so dimNSk(G ) � dimNSk(G � v)
.
The factthat there exists a null pattern ~p 2 NSk(G ) not of this typegives us a strict inequality, or in other words gives uslG (v) = �1.
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ConclusionAcknowledg-mentsReferences
A Sampling of Our Results
Proposition
If there exists ~p 2 NSk(G ) with ~p(v) 6= 0, then lG (v) = �1.
Suppose ~p 2 NSk(G ) such that ~p(v) 6= 0.
Then ev is notwinnable on G , since ev is not in the orthogonal complementof NSk(G ). It follows that a pattern on G that is null on G � vis also null at v ; otherwise that pattern would win �ev for some� 2 Z⇤
k . In particular, a null pattern on G � v extended by 0 atv is null on G , so dimNSk(G ) � dimNSk(G � v)
.
The factthat there exists a null pattern ~p 2 NSk(G ) not of this typegives us a strict inequality, or in other words gives uslG (v) = �1.
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ConclusionAcknowledg-mentsReferences
A Sampling of Our Results
Proposition
If there exists ~p 2 NSk(G ) with ~p(v) 6= 0, then lG (v) = �1.
Suppose ~p 2 NSk(G ) such that ~p(v) 6= 0. Then ev is notwinnable on G , since ev is not in the orthogonal complementof NSk(G ).
It follows that a pattern on G that is null on G � vis also null at v ; otherwise that pattern would win �ev for some� 2 Z⇤
k . In particular, a null pattern on G � v extended by 0 atv is null on G , so dimNSk(G ) � dimNSk(G � v)
.
The factthat there exists a null pattern ~p 2 NSk(G ) not of this typegives us a strict inequality, or in other words gives uslG (v) = �1.
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Lights OutIntroductionLinear Algebra
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ConclusionAcknowledg-mentsReferences
A Sampling of Our Results
Proposition
If there exists ~p 2 NSk(G ) with ~p(v) 6= 0, then lG (v) = �1.
Suppose ~p 2 NSk(G ) such that ~p(v) 6= 0. Then ev is notwinnable on G , since ev is not in the orthogonal complementof NSk(G ). It follows that a pattern on G that is null on G � vis also null at v ; otherwise that pattern would win �ev for some� 2 Z⇤
k .
In particular, a null pattern on G � v extended by 0 atv is null on G , so dimNSk(G ) � dimNSk(G � v)
.
The factthat there exists a null pattern ~p 2 NSk(G ) not of this typegives us a strict inequality, or in other words gives uslG (v) = �1.
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Lights OutIntroductionLinear Algebra
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ConclusionAcknowledg-mentsReferences
A Sampling of Our Results
Proposition
If there exists ~p 2 NSk(G ) with ~p(v) 6= 0, then lG (v) = �1.
Suppose ~p 2 NSk(G ) such that ~p(v) 6= 0. Then ev is notwinnable on G , since ev is not in the orthogonal complementof NSk(G ). It follows that a pattern on G that is null on G � vis also null at v ; otherwise that pattern would win �ev for some� 2 Z⇤
k . In particular, a null pattern on G � v extended by 0 atv is null on G , so dimNSk(G ) � dimNSk(G � v).
The factthat there exists a null pattern ~p 2 NSk(G ) not of this typegives us a strict inequality, or in other words gives uslG (v) = �1.
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Lights OutIntroductionLinear Algebra
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A Sampling of Our Results
Proposition
If there exists ~p 2 NSk(G ) with ~p(v) 6= 0, then lG (v) = �1.
Suppose ~p 2 NSk(G ) such that ~p(v) 6= 0. Then ev is notwinnable on G , since ev is not in the orthogonal complementof NSk(G ). It follows that a pattern on G that is null on G � vis also null at v ; otherwise that pattern would win �ev for some� 2 Z⇤
k . In particular, a null pattern on G � v extended by 0 atv is null on G , so dimNSk(G ) � dimNSk(G � v). The factthat there exists a null pattern ~p 2 NSk(G ) not of this typegives us a strict inequality, or in other words gives uslG (v) = �1.
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Corollary
Let G be a graph, and let v 2 V (G ). Then if lG (v) = 0,~p(v) = 0 for every ~p 2 NSk(G ).
Proof. If lG (v) = 0, then lG (v) 6= �1, so by the contrapositiveof the previous proposition, ~p(v) = 0 for every ~p 2 NSk(G ).
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Corollary
Let G be a graph, and let v 2 V (G ). Then if lG (v) = 0,~p(v) = 0 for every ~p 2 NSk(G ).
Proof. If lG (v) = 0, then lG (v) 6= �1, so by the contrapositiveof the previous proposition, ~p(v) = 0 for every ~p 2 NSk(G ).
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A Sampling of Our Results
Proposition
Let G be a graph and let v 2 V (G ). If lG (v) = 0, then for all� 2 Z⇤
k , the state �ev is winnable on G , and any winningpattern ~p for �ev satisfies p(v) 6= 0.
Proof. Suppose that lG (v) = 0. Then by the previouscorollary, ~q(v) = 0 for every ~q 2 NSk(G ). The space ofwinnable states is the orthogonal complement of the space ofnull patterns, and therefore, for all � 2 Z⇤
k , �ev is winnablebecause �ev ? ~q for all ~q 2 NSk(G ).
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Lights OutIntroductionLinear Algebra
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A Sampling of Our Results
Proposition
Let G be a graph and let v 2 V (G ). If lG (v) = 0, then for all� 2 Z⇤
k , the state �ev is winnable on G , and any winningpattern ~p for �ev satisfies p(v) 6= 0.
Proof. Suppose that lG (v) = 0.
Then by the previouscorollary, ~q(v) = 0 for every ~q 2 NSk(G ). The space ofwinnable states is the orthogonal complement of the space ofnull patterns, and therefore, for all � 2 Z⇤
k , �ev is winnablebecause �ev ? ~q for all ~q 2 NSk(G ).
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Proposition
Let G be a graph and let v 2 V (G ). If lG (v) = 0, then for all� 2 Z⇤
k , the state �ev is winnable on G , and any winningpattern ~p for �ev satisfies p(v) 6= 0.
Proof. Suppose that lG (v) = 0. Then by the previouscorollary, ~q(v) = 0 for every ~q 2 NSk(G ).
The space ofwinnable states is the orthogonal complement of the space ofnull patterns, and therefore, for all � 2 Z⇤
k , �ev is winnablebecause �ev ? ~q for all ~q 2 NSk(G ).
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Lights OutIntroductionLinear Algebra
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Proposition
Let G be a graph and let v 2 V (G ). If lG (v) = 0, then for all� 2 Z⇤
k , the state �ev is winnable on G , and any winningpattern ~p for �ev satisfies p(v) 6= 0.
Proof. Suppose that lG (v) = 0. Then by the previouscorollary, ~q(v) = 0 for every ~q 2 NSk(G ). The space ofwinnable states is the orthogonal complement of the space ofnull patterns, and therefore, for all � 2 Z⇤
k , �ev is winnablebecause �ev ? ~q for all ~q 2 NSk(G ).
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L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
A Sampling of Our Results
Proposition
Let G be a graph and let v 2 V (G ). Then if lG (v) = 0, thenfor all � 2 Z⇤
k , the state �ev is winnable on G , and any winningpattern ~p for �ev satisfies p(v) 6= 0.
Assume for the sake of contradiction that ~p is a winningpattern for �ev such that ~p(v) = 0. It follows that ~p|G�v isnull, but ~p is not null on G . We also know that for all~q 2 NSk(G ), ~q(v) = 0 and thus ~q|G�v is always a null patternon G � v . We note that ~p|G�v must be distinct from ~q|G�v forall ~q 2 NSk(G ), since the e↵ect of extending these patterns by0 at v is di↵erent. This implies thatdimNSk(G � v) > dimNSk(G ), contradicting the fact thatlG (v) = 0. Thus ~p(v) 6= 0.
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L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
A Sampling of Our Results
Proposition
Let G be a graph and let v 2 V (G ). Then if lG (v) = 0, thenfor all � 2 Z⇤
k , the state �ev is winnable on G , and any winningpattern ~p for �ev satisfies p(v) 6= 0.
Assume for the sake of contradiction that ~p is a winningpattern for �ev such that ~p(v) = 0.
It follows that ~p|G�v isnull, but ~p is not null on G . We also know that for all~q 2 NSk(G ), ~q(v) = 0 and thus ~q|G�v is always a null patternon G � v . We note that ~p|G�v must be distinct from ~q|G�v forall ~q 2 NSk(G ), since the e↵ect of extending these patterns by0 at v is di↵erent. This implies thatdimNSk(G � v) > dimNSk(G ), contradicting the fact thatlG (v) = 0. Thus ~p(v) 6= 0.
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
A Sampling of Our Results
Proposition
Let G be a graph and let v 2 V (G ). Then if lG (v) = 0, thenfor all � 2 Z⇤
k , the state �ev is winnable on G , and any winningpattern ~p for �ev satisfies p(v) 6= 0.
Assume for the sake of contradiction that ~p is a winningpattern for �ev such that ~p(v) = 0. It follows that ~p|G�v isnull, but ~p is not null on G .
We also know that for all~q 2 NSk(G ), ~q(v) = 0 and thus ~q|G�v is always a null patternon G � v . We note that ~p|G�v must be distinct from ~q|G�v forall ~q 2 NSk(G ), since the e↵ect of extending these patterns by0 at v is di↵erent. This implies thatdimNSk(G � v) > dimNSk(G ), contradicting the fact thatlG (v) = 0. Thus ~p(v) 6= 0.
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
A Sampling of Our Results
Proposition
Let G be a graph and let v 2 V (G ). Then if lG (v) = 0, thenfor all � 2 Z⇤
k , the state �ev is winnable on G , and any winningpattern ~p for �ev satisfies p(v) 6= 0.
Assume for the sake of contradiction that ~p is a winningpattern for �ev such that ~p(v) = 0. It follows that ~p|G�v isnull, but ~p is not null on G . We also know that for all~q 2 NSk(G ), ~q(v) = 0 and thus ~q|G�v is always a null patternon G � v .
We note that ~p|G�v must be distinct from ~q|G�v forall ~q 2 NSk(G ), since the e↵ect of extending these patterns by0 at v is di↵erent. This implies thatdimNSk(G � v) > dimNSk(G ), contradicting the fact thatlG (v) = 0. Thus ~p(v) 6= 0.
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
A Sampling of Our Results
Proposition
Let G be a graph and let v 2 V (G ). Then if lG (v) = 0, thenfor all � 2 Z⇤
k , the state �ev is winnable on G , and any winningpattern ~p for �ev satisfies p(v) 6= 0.
Assume for the sake of contradiction that ~p is a winningpattern for �ev such that ~p(v) = 0. It follows that ~p|G�v isnull, but ~p is not null on G . We also know that for all~q 2 NSk(G ), ~q(v) = 0 and thus ~q|G�v is always a null patternon G � v . We note that ~p|G�v must be distinct from ~q|G�v forall ~q 2 NSk(G ), since the e↵ect of extending these patterns by0 at v is di↵erent.
This implies thatdimNSk(G � v) > dimNSk(G ), contradicting the fact thatlG (v) = 0. Thus ~p(v) 6= 0.
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L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
A Sampling of Our Results
Proposition
Let G be a graph and let v 2 V (G ). Then if lG (v) = 0, thenfor all � 2 Z⇤
k , the state �ev is winnable on G , and any winningpattern ~p for �ev satisfies p(v) 6= 0.
Assume for the sake of contradiction that ~p is a winningpattern for �ev such that ~p(v) = 0. It follows that ~p|G�v isnull, but ~p is not null on G . We also know that for all~q 2 NSk(G ), ~q(v) = 0 and thus ~q|G�v is always a null patternon G � v . We note that ~p|G�v must be distinct from ~q|G�v forall ~q 2 NSk(G ), since the e↵ect of extending these patterns by0 at v is di↵erent. This implies thatdimNSk(G � v) > dimNSk(G ), contradicting the fact thatlG (v) = 0.
Thus ~p(v) 6= 0.
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
A Sampling of Our Results
Proposition
Let G be a graph and let v 2 V (G ). Then if lG (v) = 0, thenfor all � 2 Z⇤
k , the state �ev is winnable on G , and any winningpattern ~p for �ev satisfies p(v) 6= 0.
Assume for the sake of contradiction that ~p is a winningpattern for �ev such that ~p(v) = 0. It follows that ~p|G�v isnull, but ~p is not null on G . We also know that for all~q 2 NSk(G ), ~q(v) = 0 and thus ~q|G�v is always a null patternon G � v . We note that ~p|G�v must be distinct from ~q|G�v forall ~q 2 NSk(G ), since the e↵ect of extending these patterns by0 at v is di↵erent. This implies thatdimNSk(G � v) > dimNSk(G ), contradicting the fact thatlG (v) = 0. Thus ~p(v) 6= 0.
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Lights OutIntroductionLinear Algebra
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Definition (The Secondary Label of v in G )
1. Let G be a graph and suppose v 2 V (G ) with lG (v) = 0.By the previous proposition, the state ev has a winningpattern ~q, and ~q(v) 2 Z⇤
k . Let
�G (v) = �~q(v)�1 2 Z⇤k .
In this situation, all null patterns ~p on G have ~p(v) = 0,and therefore �G (v) is independent of the winning pattern~q chosen.
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Definition (The Secondary Label of v in G )
1. Let G be a graph and suppose v 2 V (G ) with lG (v) = 0.By the previous proposition, the state ev has a winningpattern ~q, and ~q(v) 2 Z⇤
k . Let
�G (v) = �~q(v)�1 2 Z⇤k .
In this situation, all null patterns ~p on G have ~p(v) = 0,and therefore �G (v) is independent of the winning pattern~q chosen.
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Definition
2. Let G be a graph and suppose v 2 V (G ) withlG (v) = �1. We define �G (v) = 0. (This secondary labelon the lG (v) = �1 vertices carries no information, but isconvenient for summation notation later.)
In either case, �G (v) 2 Z⇤k will be called the secondary label
of v . The label of a vertex v with lG (v) = 0 and secondarylabel � will typically be written as 0(�).
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Definition
2. Let G be a graph and suppose v 2 V (G ) withlG (v) = �1. We define �G (v) = 0. (This secondary labelon the lG (v) = �1 vertices carries no information, but isconvenient for summation notation later.)
In either case, �G (v) 2 Z⇤k will be called the secondary label
of v . The label of a vertex v with lG (v) = 0 and secondarylabel � will typically be written as 0(�).
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Definition (Edge Join of G1 and G2 at v and w , respectively)
Let G1 and G2 be graphs with v 2 V (G1) and w 2 V (G2). LetH = EJ({G1, v}{G2,w}) be the graph withV (H) = V (G1) [ V (G2) and E (H) = E (G1) [ E (G2) [ (v ,w).
G2G1
v w
H
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Definition (Edge Join of G1 and G2 at v and w , respectively)
Let G1 and G2 be graphs with v 2 V (G1) and w 2 V (G2). LetH = EJ({G1, v}{G2,w}) be the graph withV (H) = V (G1) [ V (G2) and E (H) = E (G1) [ E (G2) [ (v ,w).
G2G1
v w
H
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L. Ballard
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Definition (Edge Join of G1 and G2 at v and w , respectively)
Let G1 and G2 be graphs with v 2 V (G1) and w 2 V (G2). LetH = EJ({G1, v}{G2,w}) be the graph withV (H) = V (G1) [ V (G2) and E (H) = E (G1) [ E (G2) [ (v ,w).
G2G1
v w
H
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Lights OutIntroductionLinear Algebra
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Theorem
Let G1 and G2 be graphs with v 2 V (G1) and w 2 V (G2). Letdi = dimNSk(Gi ), and let H = EJ({G1, v}{G2,w}). ThendimNSk(H) is given by the following table.
lG1(v) lG2(w) dimNSk(H)
1 any d1 + d2�1 any d1 + d2 + lG2(w)� 10(�) 0(µ) d1 + d2 (µ 6= ��1)0(�) 0(µ) d1 + d2 + 1 (µ = ��1)
In particular, if lG1(v) = �1, thendimNSk(H) = dimNSk(G1 � v) + dimNSk(G2 � w).
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Vertex joining:
G1 G2
H
v1 v2
v
The following theorem will show how NSk(H) depends onlGi
(vi ) for i 2 {1, 2}.
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Vertex joining:
G1 G2
H
v1 v2
v
The following theorem will show how NSk(H) depends onlGi
(vi ) for i 2 {1, 2}.
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L. Ballard
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Vertex joining:
G1 G2
H
v1 v2
v
The following theorem will show how NSk(H) depends onlGi
(vi ) for i 2 {1, 2}.
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L. Ballard
Lights OutIntroductionLinear Algebra
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Vertex joining:
G1 G2
H
v1 v2
v
The following theorem will show how NSk(H) depends onlGi
(vi ) for i 2 {1, 2}.
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L. Ballard
Lights OutIntroductionLinear Algebra
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Theorem
For 1 i m, let Gi be a finite graph with vi 2 V (Gi ). Thegraph H = VJ({Gi , vi : 1 i m}) is defined by
V (H) =m[
i=1
V (Gi � vi ) [ {v} and
E (H) =m[
i=1
E (Gi � vi ) [((wv) : (wvi ) 2
m[
i=1
E (Gi )
).
1 If lGi(vi ) = 1 for at least one i , then lH(v) = 1.
2 lH(v) 2 {0,�1} if and only if lGi(vi ) 2 {0,�1} for all i .
Moreover, lH(v) = �1 if and only ifmX
i=1
�Gi(vi ) = m � 1( mod k).
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L. Ballard
Lights OutIntroductionLinear Algebra
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ConclusionAcknowledg-mentsReferences
A Sampling of Our Results
Theorem
For 1 i m, let Gi be a finite graph with vi 2 V (Gi ). Thegraph H = VJ({Gi , vi : 1 i m}) is defined by
V (H) =m[
i=1
V (Gi � vi ) [ {v} and
E (H) =m[
i=1
E (Gi � vi ) [((wv) : (wvi ) 2
m[
i=1
E (Gi )
).
1 If lGi(vi ) = 1 for at least one i , then lH(v) = 1.
2 lH(v) 2 {0,�1} if and only if lGi(vi ) 2 {0,�1} for all i .
Moreover, lH(v) = �1 if and only ifmX
i=1
�Gi(vi ) = m � 1( mod k).
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L. Ballard
Lights OutIntroductionLinear Algebra
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ConclusionAcknowledg-mentsReferences
A Sampling of Our Results
Theorem
For 1 i m, let Gi be a finite graph with vi 2 V (Gi ). Thegraph H = VJ({Gi , vi : 1 i m}) is defined by
V (H) =m[
i=1
V (Gi � vi ) [ {v} and
E (H) =m[
i=1
E (Gi � vi ) [((wv) : (wvi ) 2
m[
i=1
E (Gi )
).
1 If lGi(vi ) = 1 for at least one i , then lH(v) = 1.
2 lH(v) 2 {0,�1} if and only if lGi(vi ) 2 {0,�1} for all i .
Moreover, lH(v) = �1 if and only ifmX
i=1
�Gi(vi ) = m � 1( mod k).
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Lights OutIntroductionLinear Algebra
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Acknowledgments
I would like to acknowledge:
Syracuse University MGO
Dr. Darin Stephenson
Hope College
The National Science Foundation
Drs. Kristin Camenga and Rebekah Yates
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L. Ballard
Lights OutIntroductionLinear Algebra
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ConclusionAcknowledg-mentsReferences
References
Anderson, M., & Feil, T. (1998). Turning lights out with linearalgebra. Mathematics Magazine, 71(4), 300-303.
Edwards, S., Elandt, V., James, N., Johnson, K., Mitchell, Z.,& Stephenson, D. (2008). Lights out on finite graphs. 1-13.
Gi↵en, A., & Parker, D. (2009). On generalizing the“lightsout” game and a generalization of parity domination. 1-14.
Stephenson, D. R. (2012). Lights out, �+-automata, anddomination in graphs. 1-10.
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L. Ballard
Lights OutIntroductionLinear Algebra
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Theorem
Let G be a finite graph with v 2 V (G ), and let fv be the
function which extends a pattern on G � v to a pattern on
G that is zero at v . The following are equivalent:
1 lG (v) = �1,
2The state ev is not winnable on G .
3There exists p 2 Null(N(G )) with p(v) 6= 0.
4The function fv induces an injective linear
transformation from Null(N(G � v)) to Null(N(G )), anddim coker(fv ) = 1.
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L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Theorem
Let G be a finite graph with v 2 V (G ), and let fv be the
function which extends a pattern on G � v to a pattern on
G that is zero at v . The following are equivalent:
1 lG (v) = �1,
2The state ev is not winnable on G .
3There exists p 2 Null(N(G )) with p(v) 6= 0.
4The function fv induces an injective linear
transformation from Null(N(G � v)) to Null(N(G )), anddim coker(fv ) = 1.
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Theorem
Let G be a finite graph with v 2 V (G ), and let fv be the
function which extends a pattern on G � v to a pattern on
G that is zero at v . The following are equivalent:
1 lG (v) = �1,
2The state ev is not winnable on G .
3There exists p 2 Null(N(G )) with p(v) 6= 0.
4The function fv induces an injective linear
transformation from Null(N(G � v)) to Null(N(G )), anddim coker(fv ) = 1.
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L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Theorem
Let G be a finite graph with v 2 V (G ), and let fv be the
function which extends a pattern on G � v to a pattern on
G that is zero at v . The following are equivalent:
1 lG (v) = �1,
2The state ev is not winnable on G .
3There exists p 2 Null(N(G )) with p(v) 6= 0.
4The function fv induces an injective linear
transformation from Null(N(G � v)) to Null(N(G )), anddim coker(fv ) = 1.
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Lights Out onRelatedGraphs
L. Ballard
Lights OutIntroductionLinear Algebra
ResearchNull SpaceTheorems
ConclusionAcknowledg-mentsReferences
Theorem
Let G be a finite graph with v 2 V (G ), and let fv be the
function which extends a pattern on G � v to a pattern on
G that is zero at v . The following are equivalent:
1 lG (v) = �1,
2The state ev is not winnable on G .
3There exists p 2 Null(N(G )) with p(v) 6= 0.
4The function fv induces an injective linear
transformation from Null(N(G � v)) to Null(N(G )), anddim coker(fv ) = 1.
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