Graphs vs Linear algebra Euler’s Theorem on latin squares Finite geometry The bigger picture Algebraic combinatorics applied to finite geometry John Bamberg Centre for the Mathematics of Symmetry and Computation, The University of Western Australia December 1, 2011 , =2
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Graphs vs Linear algebra Euler’s Theorem on latin squares Finite geometry The bigger picture
Algebraic combinatorics applied tofinite geometry
John Bamberg
Centre for the Mathematics of Symmetry and Computation,The University of Western Australia
December 1, 2011
,
= 2
Graphs vs Linear algebra Euler’s Theorem on latin squares Finite geometry The bigger picture
Graphs vs Linear algebra
Euler’s Theorem on latin squares
Finite geometry
The bigger picture
Graphs vs Linear algebra Euler’s Theorem on latin squares Finite geometry The bigger picture
Graphs
• Vertices
• Edges: pairs of vertices (u, v)
Degree: 3
Graphs vs Linear algebra Euler’s Theorem on latin squares Finite geometry The bigger picture
Graphs
• Vertices
• Edges: pairs of vertices (u, v)
Degree: 3
Graphs vs Linear algebra Euler’s Theorem on latin squares Finite geometry The bigger picture
Good labellings
• Assign a real number to each vertex.
• For each vertex, sum the values of adjacent vertices.
• Goal: Sum at each vertex should be a common multiple of thevalue at the vertex.
?1
?
1?
1
?
1
?
1
Graphs vs Linear algebra Euler’s Theorem on latin squares Finite geometry The bigger picture
Good labellings
• Assign a real number to each vertex.
• For each vertex, sum the values of adjacent vertices.
• Goal: Sum at each vertex should be a common multiple of thevalue at the vertex.
?1
?
1?
1
?
1
?
1
Graphs vs Linear algebra Euler’s Theorem on latin squares Finite geometry The bigger picture
Good labellings
• Assign a real number to each vertex.
• For each vertex, sum the values of adjacent vertices.
• Goal: Sum at each vertex should be a common multiple of thevalue at the vertex.
?1
?
1?
1
?
1
?
1
Graphs vs Linear algebra Euler’s Theorem on latin squares Finite geometry The bigger picture
Good labellings
• Assign a real number to each vertex.
• For each vertex, sum the values of adjacent vertices.
• Goal: Sum at each vertex should be a common multiple of thevalue at the vertex.
?1
?
1?
1
?
1
?
1
Graphs vs Linear algebra Euler’s Theorem on latin squares Finite geometry The bigger picture
Good labellings
• Assign a real number to each vertex.
• For each vertex, sum the values of adjacent vertices.
• Goal: Sum at each vertex should be a common multiple of thevalue at the vertex.
−11
−1
1−1
1
−1
1
−1
1
Graphs vs Linear algebra Euler’s Theorem on latin squares Finite geometry The bigger picture
1
1
?
?
??
?
1
1 ?
Graphs vs Linear algebra Euler’s Theorem on latin squares Finite geometry The bigger picture
1
1
-23
-23
-23-23
-23
1
1 -23
Graphs vs Linear algebra Euler’s Theorem on latin squares Finite geometry The bigger picture
Linear algebra
Adjacency relation
Let V be the vertex-set of a graph.u ∼ v
Adjacency operator A on RV
Given f : V → R, we define
Af : V → R
: v 7→∑u∼v
f (u)
Graphs vs Linear algebra Euler’s Theorem on latin squares Finite geometry The bigger picture
Eigenvectors of A
11
1
1
11
1
1
1
1
−11
−1
1
−11
−1
1
−1
1
1
1
- 23
- 23
- 23
- 23
- 23
1
1 - 23
3 1 −2
Graphs vs Linear algebra Euler’s Theorem on latin squares Finite geometry The bigger picture
Algebraic graph theory
• Subject grew in the 1950’s and ‘60’s:
• Graph is regular if 1 is an eigenvector.
• #Edges = 12
∑λ2i
• #Triangles = 16
∑λ3i
• 3 distinct eigenvalues −→ strongly regular
• Smallest eigenvalue −→ independence and chromatic numbers
• Second largest eigenvalue −→ expansion and randomness properties
• Interlacing −→ substructures.
Graphs vs Linear algebra Euler’s Theorem on latin squares Finite geometry The bigger picture
Algebraic graph theory
• Subject grew in the 1950’s and ‘60’s:
• Graph is regular if 1 is an eigenvector.
• #Edges = 12
∑λ2i
• #Triangles = 16
∑λ3i
• 3 distinct eigenvalues −→ strongly regular
• Smallest eigenvalue −→ independence and chromatic numbers
• Second largest eigenvalue −→ expansion and randomness properties
• Interlacing −→ substructures.
Graphs vs Linear algebra Euler’s Theorem on latin squares Finite geometry The bigger picture
Algebraic graph theory
• Subject grew in the 1950’s and ‘60’s:
• Graph is regular if 1 is an eigenvector.
• #Edges = 12
∑λ2i
• #Triangles = 16
∑λ3i
• 3 distinct eigenvalues −→ strongly regular
• Smallest eigenvalue −→ independence and chromatic numbers
• Second largest eigenvalue −→ expansion and randomness properties
• Interlacing −→ substructures.
Graphs vs Linear algebra Euler’s Theorem on latin squares Finite geometry The bigger picture
Algebraic graph theory
• Subject grew in the 1950’s and ‘60’s:
• Graph is regular if 1 is an eigenvector.
• #Edges = 12
∑λ2i
• #Triangles = 16
∑λ3i
• 3 distinct eigenvalues −→ strongly regular
• Smallest eigenvalue −→ independence and chromatic numbers
• Second largest eigenvalue −→ expansion and randomness properties
• Interlacing −→ substructures.
Graphs vs Linear algebra Euler’s Theorem on latin squares Finite geometry The bigger picture
Algebraic graph theory
• Subject grew in the 1950’s and ‘60’s:
• Graph is regular if 1 is an eigenvector.
• #Edges = 12
∑λ2i
• #Triangles = 16
∑λ3i
• 3 distinct eigenvalues −→ strongly regular
• Smallest eigenvalue −→ independence and chromatic numbers
• Second largest eigenvalue −→ expansion and randomness properties
• Interlacing −→ substructures.
Graphs vs Linear algebra Euler’s Theorem on latin squares Finite geometry The bigger picture
Using the spectral decomposition
Spectral Theorem
The eigenspaces of A form an orthogonal decomposition of RV .
1
1
- 23
- 23
- 23- 2
3
- 23
1
1 - 23
,
−11
−1
1−1
1
−1
1
−1
1
= 0
Graphs vs Linear algebra Euler’s Theorem on latin squares Finite geometry The bigger picture
Intriguing sets
Corollary
Intriguing sets X and Y associated to different eigenvalues satisfy
|X ∩ Y | =|X ||Y ||V |
.
1
1
0
0
00
0
1
1 0
,
10
1
01
0
1
0
1
0
= 2
Graphs vs Linear algebra Euler’s Theorem on latin squares Finite geometry The bigger picture
Intriguing maps
f : V → R is intriguing ⇐⇒ for some α, β ∈ R
Af = α · f + β · 1.
(In fact, α is an eigenvalue of A.)
1
0
1
0
1
0
1
0
1
0
2
1
2
1
2
1
2
1
2
1
f
Af = f + 1
Graphs vs Linear algebra Euler’s Theorem on latin squares Finite geometry The bigger picture
Intriguing maps
f : V → R is intriguing ⇐⇒ for some α, β ∈ R
Af = α · f + β · 1.
(In fact, α is an eigenvalue of A.)
1
0
1
0
1
0
1
0
1
0
2
1
2
1
2
1
2
1
2
1
f
Af = f + 1
Graphs vs Linear algebra Euler’s Theorem on latin squares Finite geometry The bigger picture
Intriguing maps
f : V → R is intriguing ⇐⇒ for some α, β ∈ R
Af = α · f + β · 1.
(In fact, α is an eigenvalue of A.)
1
0
1
0
1
0
1
0
1
0
2
1
2
1
2
1
2
1
2
1
f
Af = f + 1
Graphs vs Linear algebra Euler’s Theorem on latin squares Finite geometry The bigger picture
Intriguing maps
f : V → R is intriguing ⇐⇒ for some α, β ∈ R
Af = α · f + β · 1.
(In fact, α is an eigenvalue of A.)
1
0
1
0
1
0
1
0
1
0
2
1
2
1
2
1
2
1
2
1
f Af = f + 1
Graphs vs Linear algebra Euler’s Theorem on latin squares Finite geometry The bigger picture
Intriguing maps
0
−1
1
0
1
0
1
0
0 0
1
3
−1
1
−1
1
−1
1
1 1
f
Af = −2 · f + 1
Graphs vs Linear algebra Euler’s Theorem on latin squares Finite geometry The bigger picture
Intriguing maps
0
−1
1
0
1
0
1
0
0 0
1
3
−1
1
−1
1
−1
1
1 1
f
Af = −2 · f + 1
Graphs vs Linear algebra Euler’s Theorem on latin squares Finite geometry The bigger picture
Intriguing maps
0
−1
1
0
1
0
1
0
0 0
1
3
−1
1
−1
1
−1
1
1 1
f Af = −2 · f + 1
Graphs vs Linear algebra Euler’s Theorem on latin squares Finite geometry The bigger picture
Af = αf f + βf 1, Ag = αg f + βg1, A1 = k1.
Corollary
Intriguing maps f and g associated to different eigenvalues satisfy