Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Realizable Graphs of Equivalence Classes
of Zero Divisors
Cathryn Holm
- St. Olaf College Work done jointly with:
Chelsea Johnson, Brigham Young University
Kaylee Kooiman, Calvin College
Brigham Young University REU 2010
Funded by NSF, DMS #0453421
September 30, 2010
Cathryn Holm - St. Olaf College -
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Outline
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Cathryn Holm - St. Olaf College -
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Background
Definition
A ring is a set of elements closed under addition and multiplication
satisfying the usual requirements for addition and the associative
and distributive properties under multiplication.
Cathryn Holm - St. Olaf College -
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Background
Definition
A ring is a set of elements closed under addition and multiplication
satisfying the usual requirements for addition and the associative
and distributive properties under multiplication.
Definition
An element a ∈ R is a zero divisor if there is some non-zero r ∈ R
such that ar = 0.
Cathryn Holm - St. Olaf College -
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Background
The zero divisor graph of R is Γ(R), the graph whose vertices are
associated to non-zero zero divisors, where there is an edge from x
to y if xy = 0.
Γ(Z2 × Z4 )
Cathryn Holm - St. Olaf College -
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Background
Definition
An ideal, I , is a subring of R such that ir ∈ I for all i ∈ I , r ∈ R
Cathryn Holm - St. Olaf College -
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Background
Definition
An ideal, I , is a subring of R such that ir ∈ I for all i ∈ I , r ∈ R
Definition
Let ann(x) = {a ∈ R | ax = 0}, called the annihilator ideal of x.
Cathryn Holm - St. Olaf College -
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Background
Define an equivalence relation by a ∼ b if and only if
ann(a) = ann(b).
Cathryn Holm - St. Olaf College -
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Background
Define an equivalence relation by a ∼ b if and only if
ann(a) = ann(b).
Graph of equivalence classes of zero divisors, ΓE (R).
Γ(Z2 × Z4 )
Cathryn Holm - St. Olaf College -
ΓE (Z2 × Z4 )
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Example: Z12
ann(2) = {6}
ann(3) = {4, 8}
ann(4) = {3, 6, 9}
ann(6) = {2, 4, 6, 8, 10}
Γ(Z12 )
Cathryn Holm - St. Olaf College -
ann(8) = {3, 6, 9}
ann(9) = {4, 8}
ann(10) = {6}
ΓE (Z12 )
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Research Question
Research Question
Out of the multiple graphs which are possible to draw, which
graphs represent the graphs of equivalence classes of zero divisors
of rings?
Cathryn Holm - St. Olaf College -
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Procedure
Examine all possible graphs on 4,5, and 6 vertices
4 vertices: 11 graphs
5 vertices: 34 graphs
6 vertices: 156 graphs
Cathryn Holm - St. Olaf College -
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Procedure
Examine all possible graphs on 4,5, and 6 vertices
4 vertices: 11 graphs
5 vertices: 34 graphs
6 vertices: 156 graphs
Eliminate unrealizable graphs by previous results
Cathryn Holm - St. Olaf College -
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Procedure
Examine all possible graphs on 4,5, and 6 vertices
4 vertices: 11 graphs
5 vertices: 34 graphs
6 vertices: 156 graphs
Eliminate unrealizable graphs by previous results
Develop methods to eliminate remaining graphs
Multiplication Class
Addition Class
Leaf Method
Cathryn Holm - St. Olaf College -
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Procedure
Examine all possible graphs on 4,5, and 6 vertices
4 vertices: 11 graphs
5 vertices: 34 graphs
6 vertices: 156 graphs
Eliminate unrealizable graphs by previous results
Develop methods to eliminate remaining graphs
Multiplication Class
Addition Class
Leaf Method
Find examples of realizable graphs
Cathryn Holm - St. Olaf College -
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Lemma (Spiroff and Wickham, 2009)
For Realizable Graphs:
1
Any two vertices of maximal degree must be adjacent.
2
ΓE (R) is connected and diam(ΓE (R)) ≤ 3.
3
If |ΓE (R)| ≥ 3, then ΓE (R) is not complete.
4
If ΓE (R) is complete r -partite, then r = 2 and ΓE (R) = K n,1 .
5
ΓE (R) is not a cycle graph.
Cathryn Holm - St. Olaf College -
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Lemma (Spiroff and Wickham, 2009)
For Realizable Graphs:
1
Any two vertices of maximal degree must be adjacent.
2
ΓE (R) is connected and diam(ΓE (R)) ≤ 3.
3
If |ΓE (R)| ≥ 3, then ΓE (R) is not complete.
4
If ΓE (R) is complete r -partite, then r = 2 and ΓE (R) = K n,1 .
5
ΓE (R) is not a cycle graph.
Cathryn Holm - St. Olaf College -
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Lemma (Spiroff and Wickham, 2009)
For Realizable Graphs:
1
Any two vertices of maximal degree must be adjacent.
2
ΓE (R) is connected and diam(ΓE (R)) ≤ 3.
3
If |ΓE (R)| ≥ 3, then ΓE (R) is not complete.
4
If ΓE (R) is complete r -partite, then r = 2 and ΓE (R) = K n,1 .
5
ΓE (R) is not a cycle graph.
Cathryn Holm - St. Olaf College -
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Lemma (Spiroff and Wickham, 2009)
For Realizable Graphs:
1
Any two vertices of maximal degree must be adjacent.
2
ΓE (R) is connected and diam(ΓE (R)) ≤ 3.
3
If |ΓE (R)| ≥ 3, then ΓE (R) is not complete.
4
If ΓE (R) is complete r -partite, then r = 2 and ΓE (R) = K n,1 .
5
ΓE (R) is not a cycle graph.
Cathryn Holm - St. Olaf College -
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Lemma (Spiroff and Wickham, 2009)
For Realizable Graphs:
1
Any two vertices of maximal degree must be adjacent.
2
ΓE (R) is connected and diam(ΓE (R)) ≤ 3.
3
If |ΓE (R)| ≥ 3, then ΓE (R) is not complete.
4
If ΓE (R) is complete r -partite, then r = 2 and ΓE (R) = K n,1 .
5
ΓE (R) is not a cycle graph.
Cathryn Holm - St. Olaf College -
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Multiplication Class Proof (-)
Proof.
Consider [vx]. Then ann([vx]) ⊇ {[w ], [y ], [z]}, but no such vertex
exists.
Cathryn Holm - St. Olaf College -
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Addition Class Proof (-)
Proof.
Consider [b + d]. Then ann([b + d]) ⊇ {[c], [f ]} but
ann([b + d]) + {[a], [e]}. No such vertex exists.
Cathryn Holm - St. Olaf College -
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Definitions
Definition
A prime ideal, P, is a proper ideal such that if ab ∈ P then a ∈ P
or b ∈ P.
Cathryn Holm - St. Olaf College -
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Definitions
Definition
A prime ideal, P, is a proper ideal such that if ab ∈ P then a ∈ P
or b ∈ P.
Definition
An associated prime, A, is a prime ideal of R such that A = ann(x)
for some x ∈ R.
Cathryn Holm - St. Olaf College -
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Leaf Proof (-)
Proof.
[a] and [e] both have leaves, hence are associated primes.
Assume [b]2 = 0 then [b]2 ∈ ann([e]).
Since [e] is a prime ideal, [b] ∈ ann([e]).
But [be] 6= 0, therefore [b]2 6= 0.
Similarly, [c]2 6= 0 and [d]2 6= 0. Thus [b] = [c] = [d].
Cathryn Holm - St. Olaf College -
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Leaf Proof (-)
Proof.
[a] and [e] both have leaves, hence are associated primes.
Assume [b]2 = 0 then [b]2 ∈ ann([e]).
Since [e] is a prime ideal, [b] ∈ ann([e]).
But [be] 6= 0, therefore [b]2 6= 0.
Similarly, [c]2 6= 0 and [d]2 6= 0. Thus [b] = [c] = [d].
Cathryn Holm - St. Olaf College -
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Leaf Proof (-)
Proof.
[a] and [e] both have leaves, hence are associated primes.
Assume [b]2 = 0 then [b]2 ∈ ann([e]).
Since [e] is a prime ideal, [b] ∈ ann([e]).
But [be] 6= 0, therefore [b]2 6= 0.
Similarly, [c]2 6= 0 and [d]2 6= 0. Thus [b] = [c] = [d].
Cathryn Holm - St. Olaf College -
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Leaf Proof (-)
Proof.
[a] and [e] both have leaves, hence are associated primes.
Assume [b]2 = 0 then [b]2 ∈ ann([e]).
Since [e] is a prime ideal, [b] ∈ ann([e]).
But [be] 6= 0, therefore [b]2 6= 0.
Similarly, [c]2 6= 0 and [d]2 6= 0. Thus [b] = [c] = [d].
Cathryn Holm - St. Olaf College -
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Leaf Proof (-)
Proof.
[a] and [e] both have leaves, hence are associated primes.
Assume [b]2 = 0 then [b]2 ∈ ann([e]).
Since [e] is a prime ideal, [b] ∈ ann([e]).
But [be] 6= 0, therefore [b]2 6= 0.
Similarly, [c]2 6= 0 and [d]2 6= 0. Thus [b] = [c] = [d].
Cathryn Holm - St. Olaf College -
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
4 Vertices
Theorem (-)
There are only 3 realizable graphs of equivalence classes of zero
divisors with 4 vertices.
Z3 × Z4
Z4 [x]/(x 2 )
Cathryn Holm - St. Olaf College -
Z[x, y ]/(x 3 , xy )
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
5 Vertices
Theorem (-)
There are only 4 realizable graphs of equivalence classes of zero
divisors with 5 vertices.
Z9 [x]/(x 2 )
Z64
Z3 [x, y ]/
(xy , x 3 , y 3 , x 2 − y 2 )
Cathryn Holm - St. Olaf College -
Z8 [x, y ]/
(x 2 , y 2 , 4x, 4y , 2xy )
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
6 Vertices
Theorem (-)
There are only 10 realizable graphs of equivalence classes of zero
divisors with 6 vertices.
Z[x, y ]/(x 4 , xy )
Z2 [x]/(x 7 )
Z27 [x, y ]/(xy , x 2 − y 2 , (3, x, y )3 )
Z[x, y ]/(x 2 , y 2 , xy , 2x, 3y )
Z27 [x, y ]/(x 2 , y 2 , (3, x, y )3 )
Cathryn Holm - St. Olaf College -
Z2 [x, y ]/(x 4 , xy , x 3 + y 2 )
Z2 [x, y , z]/(x 4 , y 2 , z 2 , x 2 y , x 2 z, xyz)
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
6 Vertices (cont.)
Theorem (-)
There are only 10 realizable graphs of equivalence classes of zero
divisors with 6 vertices.
Z2 [x, y , z]/(x 2 , y 2 , z 2 , xy )
Z2 × Z2 × Z2
Cathryn Holm - St. Olaf College -
Z2 × Z8
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Fan Question
“It is unknown to the authors whether, for each positive integer n,
the fan graph K n,1 can be realized as ΓE (R) for some ring R, or
how one would go about the general construction or argument.”
- Spiroff and Wickham, 2009
Cathryn Holm - St. Olaf College -
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Fan Graphs: 1-35
Cathryn Holm - St. Olaf College -
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Fan Graphs: 1-35
Cathryn Holm - St. Olaf College -
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Realizable Fans
Z4 [x]/(x 2 )
Cathryn Holm - St. Olaf College -
Z8 [x, y ]/(x 2 , y 2 , 4x, 4y , 2xy )
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Fan Graphs
Theorem (-)
Fan graphs on the following number of vertices are realizable:
2n ,
2n−1 + 2,
n≥3
n
2 + 1,
n
m
n
m
2
n≥2
2 − 2 + 2,
2 − 2 + 3,
n−2≥m ≥2
n−2≥m ≥2
Cathryn Holm - St. Olaf College -
n−1
+ 3,
n
m
n
m
n≥3
n≥3
2 − 2 − 2n−m + 4,
2 −2 −2
n−m
+ 5,
n−2≥m ≥2
n−2≥m ≥2
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Summary
Total Graphs
Unrealizable
By Lemma (S,W)
Our Methods (-)
Realizable
4 Vertices
11
5 Vertices
34
6 Vertices
156
8
0
3
20
10
4
74
72
10
Cathryn Holm - St. Olaf College -
Realizable Graphs of Equivalence Classes of Zero Divisors
Introduction
Background
Research Question
Unrealizable Graphs
Realizable Graphs
Fan Graphs
Summary
Acknowledgments
Acknowledgments
Advisor: Dr. Erin Martin, BYU
TA: Donald Sampson, BYU
Blake Allen, Eric New, Dane Skabelund
BYU Mathematics REU 2010
Funded by NSF, DMS #0453421
Cathryn Holm - St. Olaf College -
Realizable Graphs of Equivalence Classes of Zero Divisors