Graph families defined by their automorphisms | ||||
---|---|---|---|---|
distance-transitive | → | distance-regular | ← | strongly regular |
↓ | ||||
symmetric (arc-transitive) | ← | t-transitive, t ≥ 2 | skew-symmetric | |
↓ | ||||
_{(if connected)} vertex- and edge-transitive |
→ | edge-transitive and regular | → | edge-transitive |
↓ | ↓ | ↓ | ||
vertex-transitive | → | regular | → | _{(if bipartite)} biregular |
↑ | ||||
Cayley graph | ← | zero-symmetric | asymmetric |
In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group,^{[1]} is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cayley), and uses a specified set of generators for the group. It is a central tool in combinatorial and geometric group theory. The structure and symmetry of Cayley graphs makes them particularly good candidates for constructing expander graphs.
Let be a group and be a generating set of . The Cayley graph is an edge-colored directed graph constructed as follows:^{[2]}
Not every convention requires that generate the group. If is not a generating set for , then is disconnected and each connected component represents a coset of the subgroup generated by .
If an element of is its own inverse, then it is typically represented by an undirected edge.
The set is often assumed to be finite, especially in geometric group theory, which corresponds to being locally finite and being finitely generated.
The set is sometimes assumed to be symmetric () and not containing the group identity element. In this case, the uncolored Cayley graph can be represented as a simple undirected graph.
The group acts on itself by left multiplication (see Cayley's theorem). This may be viewed as the action of on its Cayley graph. Explicitly, an element maps a vertex to the vertex The set of edges of the Cayley graph and their color is preserved by this action: the edge is mapped to the edge , both having color . In fact, all automorphisms of the colored directed graph are of this form, so that is isomorphic to the symmetry group of .^{[note 1]}^{[note 2]}
The left multiplication action of a group on itself is simply transitive, in particular, Cayley graphs are vertex-transitive. The following is a kind of converse to this:
Sabidussi's Theorem — An (unlabeled and uncolored) directed graph is a Cayley graph of a group if and only if it admits a simply transitive action of by graph automorphisms (i.e., preserving the set of directed edges).^{[4]}
To recover the group and the generating set from the unlabeled directed graph , select a vertex and label it by the identity element of the group. Then label each vertex of by the unique element of that maps to The set of generators of that yields as the Cayley graph is the set of labels of out-neighbors of . Since is uncolored, it might have more directed graph automorphisms than the left multiplication maps, for example group automorphisms of which permute .
Main article: Schreier coset graph |
If one instead takes the vertices to be right cosets of a fixed subgroup one obtains a related construction, the Schreier coset graph, which is at the basis of coset enumeration or the Todd–Coxeter process.
Knowledge about 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. Conversely, for symmetric generating sets, the spectral and representation theory of are directly tied together: take a complete set of irreducible representations of and let with eigenvalues . Then the set of eigenvalues of is exactly where eigenvalue appears with multiplicity for each occurrence of as an eigenvalue of
The genus of a group is the minimum genus for any Cayley graph of that group.^{[6]}
For infinite groups, the coarse geometry of the Cayley graph is fundamental to geometric group theory. For a finitely generated group, this is independent of choice of finite set of generators, hence an intrinsic property of the group. This is only interesting for infinite groups: every finite group is coarsely equivalent to a point (or the trivial group), since one can choose as finite set of generators the entire group.
Formally, for a given choice of generators, one has the word metric (the natural distance on the Cayley graph), which determines a metric space. The coarse equivalence class of this space is an invariant of the group.
When , the Cayley graph is -regular, so spectral techniques may be used to analyze the expansion properties of the graph. In particular for abelian groups, the eigenvalues of the Cayley graph are more easily computable and given by with top eigenvalue equal to , so we may use Cheeger's inequality to bound the edge expansion ratio using the spectral gap.
Representation theory can be used to construct such expanding Cayley graphs, in the form of Kazhdan property (T). The following statement holds:^{[7]}
For example the group has property (T) and is generated by elementary matrices and this gives relatively explicit examples of expander graphs.
An integral graph is one whose eigenvalues are all integers. While the complete classification of integral graphs remains an open problem, the Cayley graphs of certain groups are always integral. Using previous characterizations of the spectrum of Cayley graphs, note that is integral iff the eigenvalues of are integral for every representation of .
A group is Cayley integral simple (CIS) if the connected Cayley graph is integral exactly when the symmetric generating set is the complement of a subgroup of . A result of Ahmady, Bell, and Mohar shows that all CIS groups are isomorphic to , or for primes .^{[8]} It is important that actually generates the entire group in order for the Cayley graph to be connected. (If does not generate , the Cayley graph may still be integral, but the complement of is not necessarily a subgroup.)
In the example of , the symmetric generating sets (up to graph isomorphism) are
The only subgroups of are the whole group and the trivial group, and the only symmetric generating set that produces an integral graph is the complement of the trivial group. Therefore must be a CIS group.
The proof of the complete CIS classification uses the fact that every subgroup and homomorphic image of a CIS group is also a CIS group.^{[8]}
A slightly different notion is that of a Cayley integral group , in which every symmetric subset produces an integral graph . Note that no longer has to generate the entire group.
The complete list of Cayley integral groups is given by , and the dicyclic group of order , where and is the quaternion group.^{[8]} The proof relies on two important properties of Cayley integral groups:
Given a general group , a subset is normal if is closed under conjugation by elements of (generalizing the notion of a normal subgroup), and is Eulerian if for every , the set of elements generating the cyclic group is also contained in . A 2019 result by Guo, Lytkina, Mazurov, and Revin proves that the Cayley graph is integral for any Eulerian normal subset , using purely representation theoretic techniques.^{[9]}
The proof of this result is relatively short: given an Eulerian normal subset, select pairwise nonconjugate so that is the union of the conjugacy classes . Then using the characterization of the spectrum of a Cayley graph, one can show the eigenvalues of are given by taken over irreducible characters of . Each eigenvalue in this set must be an element of for a primitive root of unity (where must be divisible by the orders of each ). Because the eigenvalues are algebraic integers, to show they are integral it suffices to show that they are rational, and it suffices to show is fixed under any automorphism of . There must be some relatively prime to such that for all , and because is both Eulerian and normal, for some . Sending bijects conjugacy classes, so and have the same size and merely permutes terms in the sum for . Therefore is fixed for all automorphisms of , so is rational and thus integral.
Consequently, if is the alternating group and is a set of permutations given by , then the Cayley graph is integral. (This solved a previously open problem from the Kourovka Notebook.) In addition when is the symmetric group and is either the set of all transpositions or the set of transpositions involving a particular element, the Cayley graph is also integral.
Cayley graphs were first considered for finite groups by Arthur Cayley in 1878.^{[2]} Max Dehn in his unpublished lectures on group theory from 1909–10 reintroduced Cayley graphs under the name Gruppenbild (group diagram), which led to the geometric group theory of today. His most important application was the solution of the word problem for the fundamental group of surfaces with genus ≥ 2, which is equivalent to the topological problem of deciding which closed curves on the surface contract to a point.^{[10]}