Cayley graph

The Cayley graph of the free group on two generators a and b

In mathematics, the Cayley graph, also known as the Cayley colour graph, is the graph that encodes the structure of a discrete group. Its definition is suggested by Cayley's theorem (named after Arthur Cayley) and uses a particular, usually finite, set of generators for the group. It is a central tool in combinatorial and geometric group theory.

Definition

Suppose that $G$ is a group and $S$ is a generating set. The Cayley graph $\Gamma =\Gamma (G,S)$ is a colored directed graph constructed as follows.

• Each element $g$ of $G$ is assigned a vertex: the vertex set $V(\Gamma )$ of $\Gamma$ is identified with $G.$
• Each generator $s$ of $S$ is assigned a color $c_{s}$.
• For any $g\in G,s\in S,$ the vertices corresponding to the elements $g$ and $gs$ are joined by a directed edge of colour $c_{s}.$ Thus the edge set $E(\Gamma )$ consists of pairs of the form $(g,gs),$ with $s\in S$ providing the color.

In geometric group theory, the set $S$ is usually assumed to be finite, symmetric, i.e. $S=S^{-1},$ and not containing the identity element of the group. In this case, the Cayley graph is an ordinary graph: its edges are not oriented and it does not contain loops.

Examples

• Suppose that G = Z is the infinite cyclic group and the set S consists of the standard generator 1 and its inverse (−1 in the additive notation) then the Cayley graph is an infinite chain.

• Similarly, if G = Z_n is the finite cyclic group of order n and the set S consists of two elements, the standard generator of G and its inverse, then the Cayley graph is the cycle C_n.

• The Cayley graph of the direct product of groups is the cartesian product of the corresponding Cayley graphs. Thus the Cayley graph of the abelian group Z² with the set of generators consisting of four elements (±1, ±1) is the infinite grid on the plane R², while for the direct product Z_n × Z_m with similar generators the Cayley graph is the n by m finite grid on a torus.

The Cayley graph of the dihedral group D₄ on two generators α and β

• The Cayley graph of the dihedral group D₄ on two generators α and β is depicted to the right. Red arrows represent left-multiplication by element α. Since element β is self-inverse, the blue lines which represent left-multiplication by element β are undirected. Therefore the graph is mixed: it has eight vertices, eight arrows, and four edges. The Cayley table of the group D₄ can be derived from the group presentation

$\langle \alpha ,\beta |\alpha ^{4}=\beta ^{2}=e,\alpha \beta =\beta \alpha ^{3}\rangle .$

• The Cayley graph of the free group on two generators a, b corresponding to the set S = {a, b, a⁻¹, b⁻¹} is depicted at the top of the article, and e represents the identity element. Travelling along an edge to the right represents right multiplication by a, while travelling along an edge upward corresponds to the multiplication by b. Since the free group has no relations, the Cayley graph has no cycles. This Cayley graph is a key ingredient in the proof of the Banach–Tarski paradox.

Characterization

The group $G$ acts on itself by the left multiplication (see Cayley's theorem). This action may be viewed as the action of $G$ on its Cayley graph. Explicitly, an element $h\in G$ maps a vertex $g\in V(\Gamma )$ to the vertex $hg\in V(\Gamma )$. The set of edges of the Cayley graph is preserved by this action: the edge $(g,gs)$ is transformed into the edge $(hg,hgs)$. The left multiplication action of any group on itself is simply transitive, in particular, the Cayley graph is vertex transitive. This leads to the following characterization of Cayley graphs:

A graph $\Gamma$ is a Cayley graph of a group $G$ if and only if it admits a simply transitive action of $G$ by graph automorphisms (i.e. preserving the set of edges).

To recover the group $G$ and the generating set $S$ from the Cayley graph $\Gamma =\Gamma (G,S)$, select a vertex $v_{1}\in V(\Gamma )$ and label it by the identity element of the group. Then label each vertex $v$ of $\Gamma$ by the unique element of $G$ that trasforms $v_{1}$ into $v.$ The set $S$ of generators of $G$ that yields $\Gamma$ as the Cayley graph is the set of labels of the vertices adjacent to the selected vertex. The generating set is finite (this is a common assumption for Cayley graphs) if and only if the graph is locally finite (i.e. each vertex is adjacent to finitely many edges).

Elementary properties

• If a member $s$ of the generating set is its own inverse, $s=s^{-1}$, then it is generally represented by an undirected edge.

• The Cayley graph $\Gamma (G,S)$ depends in an essential way on the choice of the set $S$ of generators. For example, if the generating set $S$ has $k$ elements then each vertex of the Cayley graph has $k$ incoming and $k$ outgoing directed edges. In the case of a symmetric generating set $S$ with $r$ elements, the Cayley graph is a regular graph of degree $r.$

• Cycles (or closed walks) in the Cayley graph indicate relations between the elements of $S.$ In the more elaborate construction of the Cayley complex of a group, closed paths corresponding to relations are "filled in" by polygons.

• If $f:G'\to G$ is a surjective group homomorphism and the images of the elements of the generating set $S'$ for $G'$ are distinct, then it induces a covering of graphs

${\bar {f}}:\Gamma (G',S')\to \Gamma (G,S),\quad$ where $S=f(S').$

In particular, if a group $G$ has $k$ generators, all of order different from 2, and the set $S$ consists of these generators together with their inverses, then the Cayley graph $\Gamma (G,S)$ is covered by the infinite regular tree of degree $2k$ corresponding to the free group on the same set of generators.

• A graph $\Gamma (G,S)$ can be constructed even if the set $S$ does not generate the group $G.$ However, it is disconnected and is not considered to be a Cayley graph. In this case, each connected component of the graph represents a coset of the subgroup generated by $S$.

• For any Cayley graph, considered as undirected, the vertex connectivity is equal to the degree of the graph.

Schreier coset graph

If one, instead, takes the vertices to be right cosets of a fixed subgroup $H$, one obtains a related construction, the Schreier coset graph, which is at the basis of coset enumeration or the Todd-Coxeter process.

Connection to group theory

Insights into the structure of the group can be obtained by studying the adjacency matrix of the graph and in particular applying the theorems of spectral graph theory.

See also

• Bethe lattice
• Vertex-transitive graph
• Generating set of a group
• Presentation of a group
• Lovász conjecture
• Cube-connected cycles