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 e*a = a*e = a for all a  a*(b*c) = (a*b)*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, to create networks, and to 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 Γ as follows:  The vertices of Γ are the elements of G  An edge is drawn between g and g*s for every g in G and s in S  Example: G = integers mod 4, S = {1,2}. 12 34 1 1 1 1 Fig. 2: A Cayley graph with vertices and edges labeled by group elements 22

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(a*b)=f(a)*f(b). For graphs, structure is edge connection, so f(a) is connected to f(b) if a is connected to b. 41 23 1 1 1 1 Fig. 3: An automorphism of the graph of figure 2 (rotation). 41 23 22

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 1+sqrt(3), proven using the geometric representation. For S_7, it was found to be 1+sqrt(7) using matlab.