Structural characterization of Cayley graphs

Structural characterization of Cayley graphs

Didier Caucal CNRS, LIGM, University Paris-Est, France
didier.caucal@univ-mlv.fr
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.

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