Cayley Graphs

Chloe Conlon 17337976 Grace Hanlon 17475456 Kym Murray 17414662

Introduction

Cayley graphs are graphs that are associated to a and a set of generators for that group.

Bethe Lattice Cayley Graphs Introduced by the German physicist in 1935, the Bethe • was an English mathematician. Cayley Lattice is an infinite, -free graph, with each node made an important contribution to the algebraic connected to z neighbours, where z is the coordination theory of curves and surfaces, , linear number. algebra, and elliptic functions. It is a rooted with all nodes arranged in shells • Two groups are said to be isomorphic to each other if around the origin of the lattice. they become identical after the relabelling of their The Bethe Lattice is equivalent to the Cayley graph of a elements. on n generators. • Given a group G and a generating set X, then every element in G is assigned to a vertex in X, such that there are directed edges going from a ∈ G → ax ∈ G given a colour assigned to X

Construction of a Cayley Graph Campanology Rubik’s Cube

Once we find a group that is generated by some Campanology is the art of bell ringing. The essense of a 2x2x2 Rubik’s Cube finite collection elements, we can construct a English mathematicians realised that is a Cayley graph, CG. . Thus, every group element there was a relationship between bell corresponds to an isometry. ringing (Plain Bob Minimus) and Cayley Cayley’s Mouse Trap • Here is the construction of a Cayley graph for a graphs. Since the vertices of a Cayley group G with generators a1,a2,...,an in 3 steps: graph of Sn represent all represent all • Draw one vector for every group element. possible bell ringing permutations of n Card game introduced by Cayley bells, finding a certain path (called a based on permutations of 13 cards. • For every generator aj, connect vertex g to gaj by Hamiltonian Circuit) in the graph would a directed edge from g to gaj. Label the edge with result in a change in pattern. the generator. Cayley Digraph • Repeat step 2 for every element (i.e. vertex) For example: g ∈ G. Plain Bob Minimus is a permutation from bell ringing of 1234 rounds. These Properties: are the elements of S4, the symmetric • Graph is connected. group of four elements. Example Z6 • At most one arc goes from a vertex g to vertex h • Each vertex g has exactly one arc Draw the Cayley graph for Z6, with just one generator, namely the 60 rotation. However, of each type starting at g and one we can also generate it with two generators: of each type ending at g rotations by 120 and 180. • If two different sequences of arc types staring from vertex g lead to the same vertex h then those same sequences of arc types starting from any vertex u will lead to the same vertex v

Examples of Cayley Graphs: D6, D8, D10 References

Wikipedia Britannica.com web.williams.edu Dr. Rachel Quinlan, NUIG