Cayley graphs formed by conjugate generating sets of S_n

1
Cayley graphs formed by conjugate generating sets
of S_n

• Jacob Steinhardt

2
Overview

• Cayley graphs provide a link between two
  mathematical structures -- groups and graphs.
  Given a group and some elements of the group, we
  can construct a graph.
• The purpose of my research is to explore the
  structure of this link.

3
Groups

• A group is a set S together with a mathematical
  operation such that for all a,b,c in S,
• there exists an e such that ea ae a for all
  a
• a(bc) (ab)c (associativity)?
• (a-1)a a(a-1) e for some a-1
• Examples
• Real numbers under addition
• Non-zero real numbers under multiplication
• Invertible matrices under multiplication
• One-to-one functions under function composition

4
An Important Example

• Given a finite set S, the permutations of that
  set are one-to-one functions of S onto itself.
• Example S 1,2,3, and the permutation
  3,1,2. This corresponds to the function f(1)
  3, f(2) 1, f(3) 2.
• All of the permutations together form a group,
  called the symmetric group. If S 1,2,...,n,
  then we denote the symmetric group by S_n.

5
Graphs

• A graph is a collection of vertices, along with
  edges connecting those vertices.
• Example Cities with highways between them. The
  cities are vertices, the highways are edges.
• Graphs are used in computer processors,
  tocreate networks, andto organize all of our
  information!

Fig. 1 A graph with 4 vertices and 4 edges.
6
Cayley graphs

• Given a group G and a subset S of G, we define a
  Cayley graph G as follows
• The vertices of G are the elements of G
• An edge is drawn between g and gs for every g
  in G and s in S
• Example G integers mod 4, S 1,2.

1
2
2
2
3
4
Fig. 2 A Cayley graph with vertices and edges
labeled by group elements
7
Automorphisms

• An automorphism of a group or graph is a
  permutation of the elements that preserves
  structure. One can think of it as a symmetry.
• For groups, structure is multiplication, so
  f(ab)f(a)f(b). For graphs, structure is edge
  connection, so f(a) is connected to f(b) if a
  is connected to b.

4
1
4
1
2
2
2
3
2
3
Fig. 3 An automorphism of the graph of figure 2
(rotation).
Fig. 3 An automorphism of the graph of figure 2
(rotation).
8
Prior Work

• If G is the symmetric group and S consists of
  transpositions (permutations that swap two
  elements), then the Cayley graphs are
  well-studied.
• We know that there is a strong connection between
  graph and group automorphisms in this case.
• My question what happens if we use k-cycles
  (permutations that cycle k elements)?

9
Why consider k-cycles?

• Transpositions are just 2-cycles
• 2-cycles and k-cycles have similar algebraic
  structure
• commutativity/non-commutativity
• conjugacy
• properties as generators (proved in my paper)?
• The answer For many cases, the connection
  between group and graph automorphisms is still
  there if we use k-cycles instead of
  transpositions!

10
Proof Outline

• transpositions (Yan-Quan Feng)?
• find short cycles containing a,b,ab
• then all automorphisms fixing (e) are locally
  multiplicative
• use this to lift graph automorphisms to group
  automorphisms

• k-cycles
• use short cycles tree structure to show that
  edge types are preserved through automorphism
• use this to construct longer cycles containing
  a,b,ab
• lift graph automorphisms to group automorphisms

11
Ramifications

• Cayley graphs create extremely good networks
• short travel time between vertices
• high resistance to damaged vertices and edge
• we can incorporate additional, customized
  properties into a network by choosing the right
  group
• the better we understand the connection between
  the groups and the graphs, the more control we
  have over what properties we incorporate
• automorphisms are a good place to start because
  they assign algebraic properties to a graph

12
Mathematical Program

• Many results about graphs are given in terms of
  their automorphism groups
• Generally, sparse Cayley graphs have a strong
  connection with the groups used to generate them
  dense graphs have a much weaker connection
• We don't understand very well how the
  connection weakens as graphs get denser
• k-cycles have a good graphical representation

13
Mathematical Program (cont.)?

• Goal find cases where the connection between the
  group and graph just starts to get weaker
• Analyze what happens, in terms of the much
  simpler graphical structure of the k-cycles
• This should give us clues as to how the
  automorphisms of Cayley graphs behave in general

14
Other Future Work

• For transpositions, Cayley graphs have a good
  geometric representation as a Coxeter system
  (reflections in n-dimensional space)?
• What geometric connections can we make for
  k-cycles? Can we use this to find eigenvalues?
• Ex. The second-largest eigenvalue of the Cayley
  graph for S_4 is 1sqrt(3), proven using the
  geometric representation. For S_7, it was found
  to be 1sqrt(7) using matlab.