Cayley graph

Introduction

A Cayley graph is a graph that encodes the abstract structure of a group in a visual form. Named after the British mathematician Arthur Cayley, Cayley graphs are a fundamental concept in the field of algebraic graph theory. They provide a powerful tool for visualizing and understanding the properties of groups and their elements.

Definition and Construction

A Cayley graph is constructed from a group \( G \) and a set \( S \) of generators of \( G \). The vertices of the graph correspond to the elements of the group, and edges are drawn between vertices according to the generators in \( S \). Formally, the Cayley graph \( \Gamma(G, S) \) is defined as follows:

Each element \( g \in G \) is represented as a vertex.
For each \( g \in G \) and each \( s \in S \), there is a directed edge from \( g \) to \( gs \).

If the set \( S \) is symmetric (i.e., if \( s \in S \) implies \( s^{-1} \in S \)), the Cayley graph is undirected. Otherwise, it is directed.

Examples

Cyclic Groups

Consider the cyclic group \( \mathbb{Z}_n \) of integers modulo \( n \). The Cayley graph of \( \mathbb{Z}_n \) with the generating set \( \{1\} \) is a simple cycle of \( n \) vertices. Each vertex \( i \) is connected to \( (i+1) \mod n \).

Symmetric Groups

The symmetric group \( S_n \), consisting of all permutations of \( n \) elements, has a more complex Cayley graph. For example, the Cayley graph of \( S_3 \) with the generating set \( \{(12), (23)\} \) can be visualized as a graph where each vertex represents a permutation, and edges correspond to transpositions.

Properties

Vertex Transitivity

A Cayley graph is vertex-transitive, meaning that for any two vertices \( u \) and \( v \), there exists an automorphism of the graph mapping \( u \) to \( v \). This property reflects the underlying symmetry of the group.

Diameter

The diameter of a Cayley graph, which is the greatest distance between any pair of vertices, provides insight into the structure of the group. For example, the diameter of the Cayley graph of \( \mathbb{Z}_n \) with the generating set \( \{1\} \) is \( \lfloor n/2 \rfloor \).

Growth Rate

The growth rate of a Cayley graph is related to the concept of growth of groups. It measures how the number of vertices within a given distance from a fixed vertex increases. Groups can have polynomial, exponential, or intermediate growth rates, which are reflected in their Cayley graphs.

Applications

Cayley graphs have numerous applications in various fields of mathematics and computer science. They are used in the study of group theory, combinatorics, and geometric group theory. In computer science, Cayley graphs are employed in the design of interconnection networks for parallel computing, where the vertices represent processors and the edges represent communication links.

Visual Representation

The visual representation of Cayley graphs can reveal important structural properties of groups. For example, the Cayley graph of the dihedral group \( D_n \), which represents the symmetries of a regular \( n \)-gon, can be depicted as a graph with vertices corresponding to the symmetries and edges representing the generators (rotations and reflections).

References