# Structural characterization of Cayley graphs

###### Abstract

We show that the directed labelled Cayley graphs coincide with the rooted deterministic vertex-transitive simple graphs. The Cayley graphs are also the strongly connected deterministic simple graphs of which all vertices have the same cycle language, or just the same elementary cycle language. Under the assumption of the axiom of choice, we characterize the Cayley graphs for all group subsets as the deterministic, co-deterministic, vertex-transitive simple graphs.

## 1 Introduction

A group is a basic algebraic structure that comes from the study of polynomial
equations by Galois in 1830.
To describe the structure of a group, Cayley introduced in 1878 [Ca]
the concept of graph for any group according to any generating
subset .
This is simply the set of labelled oriented edges for
every of and of .
Such a graph, called Cayley graph (or Cayley diagram), is directed and
labelled in (or an encoding of by symbols
called letters or colors).
A characterization of unlabelled and undirected Cayley graphs was given by
Sabidussi in 1958 [Sa]. These are the connected graphs whose
automorphism group has a subgroup with a free and transitive action on the
graph.
So if one wants to know whether an unlabelled and undirected graph is a Cayley
graph, we must know if we can extract a subgroup of the automorphism group
that allows to define a free and transitive action on the graph.
To better understand the structure of Cayley graphs, it is pertinent to look
for characterizations by simple graph-theoretic conditions.
This approach was clearly stated by Hamkins in 2010: Which graphs are
Cayley graphs?

Every Cayley graph is a graph with high symmetry: it is vertex-transitive
meaning that the action of its automorphism group is transitive, or
equivalently that all its vertices are isomorphic meaning that we ‘see’ the
same structure regardless of the vertex where we ‘look’.
We can characterize the Cayley graphs as vertex-transitive graphs.
By definition, any Cayley graph satisfies three basic graph properties.
First is is simple: there are no two arcs of the same source and goal.
It is also deterministic: there are no two arcs of the same source and label.
Finally the identity element is a root.
In this article, we show that these three conditions added to the condition of
being vertex-transitive characterize exactly the Cayley graphs.
This improves Sabidussi’s characterization that easily can be adapted to
directed and labelled graphs: the Cayley graphs are the deterministic rooted
simple graphs whose automorphism group has a subgroup with a free and
transitive action on the graph. In other words, we reduce this last condition
to the fact that the graph is vertex-transitive.
For such a simplification, the key result is that every strongly connected,
deterministic and co-deterministic graph is isomorphic to the canonical graph
of any of its cycle languages.
This is a fairly standard result in automata theory.

Precisely an automaton is just a directed labelled graph (finite or not) with
input and output vertices.
It recognizes the language of the labels of paths from an input to an output
vertex. Any language is recognized by its canonical automaton, namely
the automaton whose graph is the set of transitions between the (left)
residuals of , having as its unique initial vertex, and the final
vertices are the residuals of containing the empty word.
We minimize an automaton by identifying its bisimilar vertices.
Any minimal deterministic and reduced automaton is isomorphic to the
canonical automaton of its recognized language.
Moreover, any co-deterministic and reduced automaton is minimal.
The previous key result follows from these last two properties: any
deterministic and co-deterministic reduced automaton is isomorphic to the
canonical automaton of its recognized language.

An equivalent characterization of Cayley graphs is obtained by strengthening
the condition of being rooted by the strong connectivity, and simplify the
condition of being vertex-transitive by the fact that all vertices have the
same elementary cycle language.
Finally, we consider the extension of Cayley graphs for all non-empty subsets
of groups. Under the assumption of the axiom of choice, we show that these
graphs are exactly the deterministic, co-deterministic, vertex-transitive
simple graphs.

## 2 Automata

An automaton is just a directed labelled graph with input and output vertices. An accepting path is a path from an initial vertex to a final vertex. An automaton recognizes the language of accepting path labels. We recall the notions of minimal automaton of an automaton, and the notion of canonical automaton of a language. For any deterministic and reduced automaton, its minimal automaton is isomorphic to the canonical automaton of its recognized language. Moreover, any co-deterministic and co-accessible automaton is minimal. It follows that every strongly connected deterministic and co-deterministic graph is isomorphic to the canonical graph of the path language between any two vertices.

### 2.1 Definitions

We recall basic definitions for directed labelled graphs and automata.

Let be an arbitrary (finite or infinite) set of symbols.
We consider a graph as a set of directed edges labelled in .
A directed -graph is defined by a set of
vertices and a subset of
edges.
Any edge is from the source to the
goal with label , and is also written by the
transition or directly if
is clear from the context.
The sources and goals of edges form the set of
non-isolated vertices of and we denote by the set of
edge labels:

and .

Thus is the set of isolated vertices.
From now on, we assume that any graph is without isolated vertex
(i.e. ) hence the graph can be identified with its edge
set . We also exclude the empty graph .
Thus, every graph is a non-empty set of labelled edges.
As any graph is a set, there are no two edges with the same source,
goal and label. We say that a graph is simple if there are no two edges
with the same source and goal:
.
We denote by the
inverse of a graph .
A graph is deterministic if there are no two edges with the same source
and label: . A graph is co-deterministic if its inverse is deterministic:
there are no two edges with the same goal and label.
For instance, the graph
represented as follows:

is deterministic and co-deterministic. The successor relation is the unlabelled edge i.e. if for some . The accessibility relation is the reflexive and transitive closure under composition of . A graph is accessible from if for any , there is such that . A root is a vertex from which is accessible. A graph is co-accessible from if is accessible from . A graph is connected if every vertex of is a root: for all .

Recall that a word of length
over is a -tuple of letters and denoted for
simplicity i.e. is a mapping from
into associating with each position
its -th letter .
The length of a word is denoted by and for each label ,
we denote and
the number of positions or occurrences of in .
The word of length is the empty word and is denoted
by . Let be the set of words over i.e.
is the free monoid generated by for the concatenation
operation.

A language is a set of words i.e.
and
is its alphabet. For any , the language
is the left residual of
by .
For all words , is a conjugated word of .

A path of length
in a graph is a sequence
of consecutive edges, and we write
for indicating the source ,
the goal and the label word of the path.
A cycle at a vertex is a path of source and goal .
A graph is strongly connected if every vertex is a root:
for all .

The set of words labelling the paths from to of a graph
is

the path language of from to .
For the previous graph Even, the path languages are

denoted
denoted .

The cycle language at vertex is
the set of labels of
cycles at ; in particular .
We say that a (non-empty) graph

is a circular graph if for all

and in that case, we denote by this common cycle language.
In other words, a graph is circular if we read the same cycle labels from any
vertex. The graph Even is circular.
Every acyclic graph is circular and of language
.

The path relation of a deterministic graph is a residual operation for
recognized languages.
{lemma}
For any and with
deterministic, .
An automaton is a graph with a subset
of initial vertices and a subset
of final vertices.
The language recognized by is

.

An automaton is accessible (resp.
co-accessible) if is accessible from (resp.
co-accessible from ); is reduced if it is
accessible and co-accessible.
We say that is deterministic if is
deterministic and .
Similarly is co-deterministic if is
co-deterministic and .
Two automata and are equivalent if they
recognize the same language: .

### 2.2 Minimal automata

We reduce an automaton by identifying bisimilar vertices.
For any deterministic co-accessible automaton, the bisimulation coincides with
Nerode’s congruence.
Any co-deterministic and co-accessible automaton is minimal and its
determinization remains minimal.

Let us consider automata and
.

A simulation from into is a relation
such that

we say that is simulated by and then any word recognized by is recognized by :

if is simulated by then .

A morphism from into is a mapping from into which is a simulation:

and and .

A simulation from into whose the inverse
relation is also a simulation is a bisimulation from on
and we say that and are
bisimilar ; in this case, they recognize the same language.
A bisimulation of is a bisimulation from on
.

A reduction from into is a mapping
from into which is a bisimulation, and we write
or directly
if we do not specify a reduction.
Thus, a reduction is a morphism whose inverse relation is a bisimulation.

Therefore two automata are bisimilar if and only if they are reducible into a
same automaton.

An injective reduction from into is an
isomorphism and we write
or directly .

A congruence of is an equivalence on which
is a bisimulation of .

The quotient of by a congruence is the
automaton
with

and for any .

which is reductible from : . Thus and its quotient under a congruence recognize the same language. The family BiSim of bisimulations of is closed under arbitrary union, inverse and composition. Let

be the greatest bisimulation of which is also the greatest
congruence of .

The minimal automaton Min of is the
quotient of under its greatest bisimulation :

.

Therefore two automata are bisimilar if and only if their minimal automata are
isomorphic.

An automaton is minimal if is the identity
i.e. is isomorphic to .

For deterministic and co-accessible, its greatest bisimulation
is Nerode’s congruence [Ne].
{lemma}
For any co-accessible automaton with
deterministic,

for all .
For any graph , we denote by the set of vertices accessible from a vertex in
by a path in labelled by .

We determinize any automaton into the following
automaton :

which is deterministic, accessible and recognizes .
Moreover is co-accessible when is
co-accessible, and minimal if in addition is co-deterministic.
{lemma}
For any automaton co-deterministic and co-accessible,

and are minimal.
For any automaton , its inverse
recognizes the mirrors of the words of
.
We co-determinize into the equivalent automaton
which is
co-deterministic and co-accessible.
Lemma 2.2 provides a fairly standard transformation of any
automaton into a deterministic minimal equivalent automaton: we apply the
co-determinization followed by the determinization.
{proposition}
For any automaton , the automaton
is minimal,

deterministic, reduced and recognizes .

### 2.3 Canonical automata

For any language and up to isomorphism, there is a unique minimal,
deterministic and reduced automaton recognizing . Such an automaton is
given by the residual graph of with the unique initial vertex
and the final vertices are the residuals of containing the empty
word. Any reduced, deterministic and co-deterministic automaton
is isomorphic to the canonical automaton of the language recognized by
.

To every language is associated its canonical graph or
residual graph:

.

For instance and . Thus is the following graph which is isomorphic to the graph :

The canonical automaton of any language is the automaton

which is the unique minimal, deterministic and reduced automaton recognizing
.
{lemma}
For any deterministic and reduced automaton , the automaton
is

isomorphic to .
This Lemma 2.3 restricted to finite automata is the Myhill-Nerode
theorem [Ha, HU].
Lemma 2.3 with Proposition 2.2 (or Lemma 2.2)
imply the isomorphism of equivalent automata which are reduced, deterministic
and co-deterministic.
{proposition}
For any automaton reduced, deterministic and co-deterministic,
is

isomorphic to .
We just see that the graph Even is isomorphic to
or
.
This generalizes to any strongly connected, deterministic and co-deterministic
graph by applying Proposition 2.3 to the automaton
for every vertices .
{corollary}
For any graph strongly connected, deterministic and co-deterministic,

is isomorphic to
for all .
This corollary implies that any strongly connected, deterministic and
co-deterministic graph is minimal with respect to any of its cycle languages.
It follows from Corollary 2.3 that two strongly connected
deterministic and co-deterministic graphs are isomorphic if they have a same
path language.
{corollary}
For any graphs strongly connected, deterministic and
co-deterministic,

if for some
and then .
This corollary is a key property to provide a structural characterization of
Cayley graphs.

## 3 Cayley graphs

Sabidussi’s theorem characterizes the undirected and unlabelled Cayley graphs as the connected graphs having a free transitive action by a subgroup of the automorphism group. We simply adapt this theorem to directed labelled graphs by replacing the connectedness with the conditions of being rooted, deterministic and simple.

### 3.1 Cayley graphs and Sabidussi’s theorem

Let be a group i.e. a set with
an associative internal binary operation such that there exists an
identity element and each has an
inverse .
Let be a non-empty
generating subset of : for any , there
are and such that
.
Let be an injective mapping
coding each by an element .
The image of [[]] is the set
of labels of .

The Cayley graph of is the
graph

.

This graph is deterministic, co-deterministic, simple and strongly connected:

for all
and .

From this path, we deduce that any Cayley graph is circular and of language

.

By Corollary 2.3, is
isomorphic to the canonical graph
.

A well-known characterization of the unlabelled and non-oriented Cayley graphs
was given by Sabidussi [Sa]. Let us recall this characterization.

First of all, a left action of on a set is a
mapping associating to
each the image such
that for all and ,

and

Note that for any , the mapping
is a permutation of
.
Thus, a group action of on may be seen as a group
homomorphism from into the group of permutations of .
We say that the action is transitive if

for all , there exists such that
.

We also say that the action is free if

for all , if there exists such that
then .

So a free and transitive action means that

for all , there exists a unique such that
.

Let be a (directed and labelled) graph.

An action of on is an action of
on which is a morphism of i.e.

for all ,
, .

Therefore, a group action of on may be seen as a group
homomorphism from into the group of
automorphisms of i.e. of isomorphisms from to
.

We say that vertices of a graph are isomorphic and
we write if there is an automorphism of
such that .

A graph is vertex-transitive if there exists a transitive group
action on . This means that acts transitively on
or equivalently that all its vertices are isomorphic: for
all . In particular, any vertex-transitive graph is circular.

First, we adapt to all Cayley graphs a Sabidussi’s characterization for a
given group.
{proposition}
A graph is isomorphic to a Cayley graph of a group if
and only if

is a deterministic rooted simple graph with a free transitive action of
on .

###### Proof.

: Assume that
for some
generating subset of and some coding
[[]] of .
The vertex set of is
whose group operation is a free transitive action
of on .

In particular is vertex-transitive
for any subset of .

: Let be a free transitive action of
on .

Let us check that is isomorphic to a Cayley graph of
by simply adapting the proof of the sufficient condition of Sabidussi’s
theorem.

As is rooted, we can pick a root of .

For all there is a unique such that
.

Thus for all ;
in particular . We define

.

As is simple and deterministic, we define the following injection
[[]] from into by

for any .

By renaming each vertex of by , we get
isomorphic to the graph

.

We check (see Appendix) that
and
is a generating subset of .
∎

The determinism condition of this proposition is necessary.
For instance, the following simple and strongly connected graph :

is not deterministic hence is not a Cayley graph, while this graph has a free
transitive action of the (cyclic) group of order .

Proposition 3.1 is a restricted characterization of Cayley
graphs since it is relative to a group .
We say that a graph has a free transitive action if there
exists a group with a free transitive action
on ; in that case
is a subgroup of
and its canonical action
is free and transitive on .
Proposition 3.1 give a simple generalization of
Sabidussi’theorem to labelled directed graphs.
{proposition}
A graph is a Cayley graph if and only if is deterministic,
rooted, simple, with a free transitive action.
Proposition 3.1 characterizes the Cayley graphs using two
conditions of different nature. The first condition is structural: the graph
must be deterministic, simple and rooted. The second condition is algebraic
namely the existence of a free transitive action.
We now give a characterization that is only structural by restricting the
algebraic condition to the vertex-transitivity: we no longer need to extract a
subgroup of the automorphism group whose the canonical action is free and
transitive.

### 3.2 Cayley graphs of languages

We briefly recall the definition of a group by a language whose letters form
the set of generators and the words define the set of relators [MKS].
Let be a language.

The word operation of deleting a word of is the rewriting
according to :

for any and

and the inverse operation is the insertion of a word
of .

The derivation and the Thue congruence
of are the reflexive and transitive closure
under composition of respectively and
.

The equivalence class of with respect to
is denoted
.

We say that a language is a group presentation language if

i.e. and
for all , there exists such that
for all .

In that case, the quotient of
under the congruence is by a group
for the operation

for all

and we define as being the Cayley graph of
generated by the subset
:

where is encoded by for
all ; this makes sense from and this graph is
non-empty by .

For instance is represented by
the following tiling plane:

where every simple (resp. double) arrow is labelled by (resp. ), and is

As is a Cayley graph, it is circular and its language is
.
{lemma}
For any group presentation language ,
and
.
We say that is a stable language if

for any

meaning that is preserved by insertion and deletion of factor in .
By iterating these two word operations from , it only gets all
words of .
{lemma}
A non-empty language is stable if and only if
.

## 4 Graph characterizations of Cayley graphs

We begin with basic graph properties, especially for circular graphs. We then give a first characterization of Cayley graphs: they are the vertex-transitive and rooted deterministic simple graphs (Theorem 4.1). They are also the circular and strongly connected deterministic simple graphs (Theorem LABEL:MainBis). We can also replace the circularity by the elementary circularity because every vertex-transitive graph is elementary circular which is then circular (Lemma LABEL:CircEleCirc). Another significant characterization concerns Cayley graphs for all subsets of groups: under ZFC, they are the deterministic, co-deterministic, vertex-transitive simple graphs (Theorem LABEL:MainFour).

### 4.1 A first graph characterization

We consider the family of deterministic, rooted, simple graphs
which are vertex-transitive. We want to establish that these graphs are Cayley
graphs. In particular, any graph of should be strongly
connected.
{lemma}
Any rooted vertex-transitive graph is strongly connected.
Furthermore any graph of should be co-deterministic.
{lemma}
Any deterministic and strongly connected circular graph is co-deterministic.
By Corollary 2.3, Lemmas 4.1 and 4.1,
any graph of is isomorphic to the canonical graph of its
path languages, hence in particular to .
This cycle language is stable.
{lemma}
For any deterministic circular graph , is a stable
language.
For any circular graph, the cycle language is closed under conjugacy, and any
label of the graph is a letter of this language when the graph is strongly
connected.
{lemma}
For any strongly connected circular graph ,

is closed under conjugacy and of letter set
.
Let us give a condition on a circular graph for its cycles to form a group
presentation language.
{lemma}
For any strongly connected, deterministic circular simple graph ,

is a group presentation language.
Any vertex-transitive graph is circular.
By Lemma 4.1 and Corollary 2.3, the converse is true when
the graph is deterministic and strongly connected.
{lemma}
For any deterministic and strongly connected graph ,

is vertex-transitive if and only if is circular.
We are able to establish a first structural characterization of Cayley graphs.
{theorem}
A graph is a Cayley graph if and only if it is deterministic, rooted, simple

and vertex-transitive.