Cayley graphs
Introduction
A Cayley graph, named after British mathematician Arthur Cayley, is a graphical representation of a group, with a vertex for each group element and an edge for each operation that connects two elements. It is a concrete, visual way to understand the structure of a group, and it is a fundamental concept in group theory, a branch of abstract algebra.
Definition
Formally, a Cayley graph is defined as follows: Given a group G with a set S of generators, the Cayley graph Γ = Γ(G, S) is a directed graph where:
1. Each element g of G is represented by a vertex. 2. For each g in G and each s in S, there is a directed edge from g to gs.
This definition assumes that the group operation is performed on the right and that the generators are not their own inverses. If the operation is performed on the left or if some generators are their own inverses, the definition can be modified accordingly.
Properties
Cayley graphs have several important properties that make them useful in the study of groups:
1. Connectedness: If the set of generators S is chosen such that it generates the group G, then the Cayley graph Γ(G, S) is connected. This means there is a path from any vertex to any other vertex. 2. Regular: Every Cayley graph is a regular graph, meaning each vertex has the same degree (the same number of edges meeting at the vertex). This degree is equal to the number of generators. 3. Vertex-transitive: Cayley graphs are vertex-transitive, meaning there is an automorphism (a graph isomorphism from the graph to itself) that maps any vertex to any other vertex. This reflects the fact that all elements of a group are "equal" in the sense that they play the same role in the group's structure. 4. Cayley's Theorem: Every group G is isomorphic to a subgroup of the symmetric group acting on G. This can be visualized using a Cayley graph.
Applications
Cayley graphs are used in many areas of mathematics and computer science:
1. Group Theory: Cayley graphs provide a way to visualize groups, which can make abstract concepts more concrete and intuitive. 2. Graph Theory: Cayley graphs are a special class of graphs with interesting properties, and studying them can lead to insights about other graphs. 3. Computer Science: Cayley graphs are used in the design of networks and algorithms, particularly in distributed computing where the nodes of the network form a group under some operation. 4. Chemistry: In crystallography, Cayley graphs can represent the structure of a crystal, where the vertices represent atoms and the edges represent bonds.