Automorphisms of Cayley graphs

Abh. Math. Sere. Univ. Hamburg 61 (1991), 73-81

Automorphisms of Cayley Graphs

By U. BAUMANN

Abstract. The aim of this paper is to characterize subgroups of non-regular automorphism groups of Cayley graphs. Relations between group automorphisms, graph automorpNsms and permu~atior~s of the edge set of Cayley graphs are investigated.

Introduction

The study of relations between groups and graphs is an interesting topic of algebraic graph theory. Since FRUCHT'S result [4] properties of automorphism groups of graphs have been studied intensively. Cayley graphs play an essential role in constructing graphs with a prescribed abstract automorphism group. BABAI [1] initiated the investigation of properties of Cayley graphs.

Until now an algebraic characterization of automorphism groups of arbitrary Cayley graphs is not known. But the problem of 'graphical regular representation' of finite groups is solved: IMRICH [9] characterized the abelian groups which can be represented by Cayley graphs with regular automorphism groups. HETZEL [8] determined all solvable groups without a graphical regular representation, and GODSIL proved [6] that every non-solvable group admits a graphical regular representation.

Less is known on non-regular automorphism groups of Cayley graphs. Since Cayley graphs can be completely described by groups and their generating sets it may be expected that there are close relations between automorphisms of groups and automorphisms of Cayley graphs. Such relations are the topic of this paper.

Concepts and notation not defined in this paper will be used as in HARARY [7] and WIELANDT [t2].

Let H be a finite group and w a generating set of H satisfying w -t = w and not containing the identity element. The vertices of the Cayley graph G(H, w) are the elements of H, and two vertices hi, hj determine an edge {hi, hj} iff h71hj E w. Now G(H, w) is a finite simple undirected connected graph.

Automorphisms of a graph are adjacency preserving permutations of the vertex set. It is easy to see that every element hi of H defines an automorphism

hi " h , hih (h E H)

of G(H, w). These automorphisms hi constitute a subgroup H of the automorphism group of the graph. The permutation group H is a regular one on the

74 U. Baumann

set of elements of the group H, i.e. only the identity element of H has fixed points and for any two elements hi, h / o f H there is (exactly one) element of

mapping hi to hj. Conversely, it is known [2] that every connected graph G having an

automorphism group with a regular subgroup S is isomorphic with a Cayley graph: The neighbours vl, v2 . . . . . vr of an arbitrary given vertex v of G define elements si (i = 1,2 . . . . . r) of S by vi = si(v). Simple verifications show that G is isomorphic with G(S, {Sl,S2 . . . . . s,}), where f :si(v) ~ sF 1 (si E S) is an isomorphism.

Often it is useful to consider a second Cayley graph isomorphic with G: Each element si of S determines a permutation

zi : s (v ) - - - , sis(v) (s ~ S).

Now the equalities szi = zis (s E S) hold. Moreover, the centralizer of the regular permutation group S is also regular [12]. Hence the elements zi constitute the centralizer Z of S in the symmetric group, and

f : s i , z i -1 (s iES)

is an isomorphism. The graph G is isomorphic with G(Z, {z? 1, Z21 . . . . . Z r 1 }).

Graph Automorphisms and Group Automorphisms

FRUCHT stated [5] that, if w is a generating set of a group H invariant under a group automorphism rc of H then rc is an automorphism of the Cayley graph G(H, w).

The consideration of group automorphisms of H mapping w onto itself makes it possible to characterize a subgroup of the automorphism group of G(H, w) which contains these group automorphisms as a proper subgroup. Before formulating this result for all graphs isomorphic with Cayley graphs, some notations and a new concept are needed.

Let G always denote a graph whose automorphism group A(G) contains a regular subgroup S. The normalizer of S in the symmetric group shall be denoted by S N, i.e. a permutation p of the vertices of G is an element of S N iff p- lSp = S. If a is an automorphism of G such that a group automorphism

of S satisfies

a' : si ~ s'i (sg ~ S)

a :si(u) , s i (v) ( s i t S )

for suitable vertices u, v, then the graph automorphism a is said to be induced by the group automorphism a t.

Proposition 1. A(G) N S N is the group of all automorphisms of the graph G which are induced by group automorphisms of S.

Automorphisms of Cayley Graphs 75

Proof. Let a be an element of A(G) A S N. Then

a :si(v) , a-lsia(a(v)) (si E S)

for an arbitrary fixed vertex v of G. Now a E S n implies that

a' : Si --'--* a - l s i a (si E S)

is an automorphism of S. Therefore the graph automorphism a is induced by the group automorphism a' of S.

Suppose, an automorphism

a' : S i ~' St i (S i C S )

of the group S induces an automorphism

a :si(u) , s'i(v) (si E S)

of the graph G. Let sy be an element of S. Then

a - l s j a : a ( s i (u ) ) , a ( s i s j (u ) ) (si E S )

holds. Now a' is a group automorphism and therefore

a- l s ja : s'i(v ) ~ s'isj(l) ) (s' i ~_ S )

is equal to the element s~ of S. Hence a c A(G) A S N. []

Let aha2 be graph automorphisms of G which are induced by group automorphisms

a'l : si , s' i (si E S) and a' 2 : s I ' dii (s'i E S ) ,

respectively. Then there are vertices u, v, w such that

al :si(u) , s'i(v) (si ~ S)

Hence the graph automorphism

ala2 : s/(u)

and a2 : s'i(v) , s'i(w) (s' i E S) .

, ~/(w) (s~ e S)

is induced by the group automorphism a'La' 2. Now it is easy to see that a set U' of group automorphisms of S constitutes

a subgroup of the automorphism group of S iff the graph automorphisms of G induced by the elements of U' constitute a subgroup of A(G).

The inner automorphisms of S induce the elements of Z S , since

-~ : si(u) , s71sisj(v) (si ~ S)

is a product psi, where an element sa of S satisfies v = &(u) and hence

p : ~(u) ~ (sksTl)si(u) (s~ ~ S)

is an element of the centralizer Z of S in the symmetric group.

76 U. Baumann

Permutations of the Edge Set Generated by Graph Automorphisms

The consideration of permutations of the edge set generated by graph automorphisms is a suitable mean for the characterization of subgroups of the automorphism group of the graph. For this purpose, the edge set of any graph G with a regular subgroup S of its automorphism group can be partitioned into pairwise disjoint subsets

{s{~,sj(,)} : s �9 s } ,

where v is an arbitrary fixed vertex and the permutations sj are suitable elements of S.

Now a set {s{v, sj(v)} : s E S} with sj E S which is not a subset of the edge set of G contains no edges of this graph. The following theorem characterizes all graph automorphisms which generate permutations of the sets

{s(v, sj(v)) : s ~ S} (s: �9 s )

of ordered pairs, i.e. which map each of these sets onto itself or onto another of these sets.

Theorem 1. A(G) A S N is the group o f all automorphisms o f the graph G which generate permutations o f the classes {s(v, sj(v)) : s E S} (sj c S ).

Proof Let a be an element of A(G)NS N. Then there are a group automorphism

a' :si ,s'i ( s i � 9

and a vertex w such that

a :si(v) , s'i(w) (si �9 S).

Now a maps each set {sl(v, si(v)) :sl ~ S}

onto a set {s'~(w,s~(w)) . s'~ �9 s } .

This image contains a pair (V, Sk(V)), and therefore

{s'~(w,s~w)) : s; e s} = {s(v, sk(~)) : s e s } .

Hence the automorphism a generates a permutation of the classes considered. Suppose, an automorphism a of G generates a permutation of the classes

{s(v, sj(v)) : s E S} (sj ~ S). Then

a :si(v) , a-lsia{a(v)) (si E S)

holds, and the permutations a-lsia (s~ ~ S) map each class {s(v, sj(v)) : s E S} onto itself. It is obvious that the permutations generating the identical


permutation of the classes {s(v, sj(v)) �9 s ~ S} (sj E S) are uniquely determined by the image of one vertex. Thus they are contained in S. Therefore

a' �9 si , a-lsi a (S i E S)

is a group automorphism of S inducing the automorphism a of G, and a is an element of A(G) A S N. []

The group A(G) n S u can also be characterized in the following way: The centralizer Z of the regular group S in the symmetric group is a regular permutation group, too. Now the normalizers S u and Z u of S and Z, respectively, in the symmetric group coincide, and therefore

A(G) A S N = A(G) N Z N .

It can be proved analogously to Proposition 1 that A(G) N Z s is the group of all automorphisms of the graph G which are induced by group automorphisms of Z. The elements zj of Z determine classes

{z(v, zj(v)) ' z ~ z } .

(It is possible that such a class contains both edges and non-edges of G.) These facts imply

Corollary 1. An automorphism o f G generates a permutation o f the classes {s(v, sj(v)) " s E S} (sj E S ) iff it generates a permutation o f the classes {=(v, z j (v)) " z ~ z} (=j ~ z ). []

Note that

z j �9 si(v) , s js i (v) (s~ e S) ( j = 1 ,2 . . . . . n)

are the elements of Z. Hence

{z~(~,zj(~)) . ~ e z } = { ( z ~ ( v ) , z i ( @ @ ) : z~ e z }

= {(z~(~),~i(sj(@) ~ ~ z}

= {(si(v) ,sisj(v)) "si ~ s }

= {(si(v) ,sj(si(v))) "s~ ~ s }

= {(w, sj(w)) " w ~ v(@}.

The classes {s(v, sj(v)) " s E S} and {(w, sj(w)) " w e V(G)} are equal iff sj is an element of the centre of the group S.

Subgroups of the Automorphism Group of Saturated Graphs

It was already mentioned above that, for any graph G under consideration there is an isomorphic graph G which can be assigned to the centralizer Z of

78 U. Baumann

the group S in the symmetric group. The edge set of G can be partitioned into pairwise disjoint subsets {z{v, zj(v)} " z E Z}, where v is an arbitrary fixed vertex and the permutations zj are suitable elements of Z.

Now G is called a saturated graph iff G = G. This means, for every saturated graph G the class {s{v, sj(v)} s c S} is a

subset of the edge set of G iff {{v, sj(v)} �9 v �9 V(G)} is also a subset of the edge set.

In this section relations between subgroups of the automorphism group of saturated graphs shall be under consideration. The groups S and Z are special subgroups. The elements of these groups are enumerated as defined above, i.e.

zj " si(v) , sjsi(v) (si E S) (j = 1,2 . . . . . n).

Moreover, the permutation

a " si(v) , zF l (v ) (si �9 S)

maps a subset {sdv, sj(v)} .s, �9 s}

of the edge set of a saturated graph G onto

{z; - ' {v , z71(v)} . z ? ' �9 z}

(note that f �9 si ~ z[ -1 (si E S) is an isomorphism), where this image is also a set of edges. Thus ~ is an automorphism of G. Hence, for each subgroup U of the automorphism group of a saturated graph G there is a conjugated subgroup ~ - I u ] . This conjugated subgroup shall be denoted by U.

Corollary 2. Let U be a subgroup o f the automorphism group A(G) o f a satu-

rated graph G. Then U = U.

Proo f By definition, U = ~-1U~ = ~-2U-a.

Furthermore, a 2 maps each vertex si(v) onto

-a(z; -1 (v)) = -a(s? 1 (v)) = zi(v) = s~(v) .

Hence the order of a is 2, and U = U. []

Corollary 3. For any saturated graph G, S = Z and Z = S.

Proof. By Corollary 2, it is sufficient to prove S = Z. Let sj be an arbitrary element of S. Then

a-lsja :a(si(v)) , ~(sj(s~(v))) (s~ �9 s ) ,

i.e. -a-lsj-a : z? l ( / ) ) ~' zT l (z i - l ( / ) ) ) (zi �9 Z) ,

and a-~s)a is an element of Z. Therefore a-~Sa = S = Z. []


If G is a saturated graph, then ~ is an element of the normalizer of the group A ( G ) n S N in the automorphism group of G:

Corollary 4. For any saturated graph G,

A(G) A S N = A(G) N S N .

Proof. If a is an element of A(G) n S N, then there are a group automorphism

a' " si ~ st(i) (siES)

and a vertex Sk(V) such that

a "si(v) , sttO(Sk(V)) (si ~- S).

Now -d-la-d " z[-l(v) ~ zffil(z~l(v)) (zi e Z)

is an automorphism of G induced by the group automorphism

a r t - - 1 �9 Zi 1 ~ Zt(i) (Z i E Z )

of Z. As it was stated above, A(G) N S N is the group of all automorphisms of

the graph G which are induced by group automorphisms of Z. Hence

A(G) n S N = a-l(A(G) O SN)fi = A(G) n S N . []

There are further pairs U, U of subgroups of automorphism groups of saturated graphs, which can be characterized by characteristic properties of generated permutations of the edge set. Especially, let As(G) denote the group of all automorphisms of G which map each class

{s{v, sj(v)} "s s} (sjc s)

onto itself. Moreover, let A~(G) be the normalizer o f AAG) in the symmetric group�9 The following theorem holds not only for saturated graphs�9

Theorem 2. A(G) n ANs (G) is the group of all automorphisms of the graph G which generate permutations o f the classes {s{v, sj(v)} " s E S} (sj E S).

Proof. Let a be an element of A(G) n ANs (G), and let

{a(x),a(y)}, {a(x'),a(y')}

be two elements of the image of {s{v, sj(v)} �9 s c S} under the action of a. It is obvious that there is an clement a~ of S mapping the element {x,y} of {s{v, sj(v)} : s c S} onto the element {x',y'} of this set. Now a-la~a maps {a(x),a(y)} onto {a(x'),a(y')}. Since S is a subgroup of As(G), as is an

80 U. Baumann

element of S and a is an element of A(G) r AN(G), the permutation a-la is an automorphism of the graph G contained in As(G). Hence the image {s{v, sj(v)} : s �9 S} under the action of a is a set {s{v,&(v)} : s �9 S}, and tl automorphism a generates a permutation of the classes {s{v, sj(v)} : s �9 (sj �9 s).

Suppose, an automorphism a of G permutes the classes {s{v, sj(v)} : s S} (sj �9 S). For any element as of As(G), the permutation a-lasa is automorphism of G mapping each set {s{v, sj(v)} : s �9 S} onto a set

asa({s{v,s)(v)}:s �9 S}).

The permutation as maps {s{v, s}(v)} "s �9 S} onto itself, and a maps this s

onto {s{v, sj(v)} : s �9 S}. Now a-lasa is an element of As(G). Therefore a contained in A(G) Cl AUs (G).

Now a characteristic property of the group

A(G) f? ANs (G)

can be stated. In general, this group is not equal to A(G) C~ AN(G). Equali not even holds for all saturated graphs.

Theorem 3. For any saturated graph G, A(G) N ANs (G) is the group of all aut morphisms of the graph G which generate permutations of the classes

Proof. By

a permutation

{z{v, zj(v)} : z �9 z } (zj �9 z) .

A(G) A A~ (G) = -a -1 (A(G) N AsU(G))E,

a :si(v) , st(i)(v) (si �9 S)

is contained in A(G) N AN(G) iff the permutation

a - l a a : Z/-I(v) ' Z~il(V) (Zi �9 Z )

is an element of A(G) N AN(G). Therefore a permutes the classes

{si{v,, say)} :s c s}

iff ~- la~ permutes the classes

{ z ; - l {v , z? l (v ) } : z? ~ �9 z}

Note that

(sjc S)

(Zj -1 E Z) .

{Z/~I{/),ZTI(/))} : Z~ 1 E Z } : {Zi{V, Zj(V)} : Z i E Z }

for every zj E Z. Since G is a saturated graph, the permutation a is an automorphism of

iff ~-la~ is an automorphism of this graph. Now the assertion follows fro Theorem 2.


Especially note that A,(G) is the group of all automorphisms of a saturated graph G which map each class

{z{v, zAv)} :z z} (zj z)

onto itself. Investigating the structure of the groups As(G) and As(G) one can state

the interesting fact that the centres of these groups coincide. The group S is a subgroup of As(G), the centralizer Z of S is contained in As(G), and the centre of the group As(G) and A~(G), respectively, is a subgroup of S ~ Z.

The results stated in this paper can also be transfered to Cayley graphs of infinite groups.

References

[1] L. BABAI, On the Abstract Group of Automorphisms, London Math. Soc. Lecture Note Series 52, Combinatorics, Cambridge-London 1981, 140.

[2] N.L. BIGGS, Algebraic Graph Theory, Cambridge 1974.

[3] P.J. CAMERON, Automorphism Groups of Graphs, Graph Theory 2 (1983), 89-127.

[4] R. FRUCHT, Herstellung von Graphen mit vorgegebener abstrakter Gruppe, Com- positio Math. 6 (1938), 23%250.

[5] R. FRUCHT, A Non-regular Graph of Degree Three, Can. J. Math. 4 (1951), 240-247.

[6] C.D. GODStL, GRR's for Non-solvable Groups, Algebraic Methods in Combi- natorics (Proc. Conf. Szeged, L.Lovasz et al. eds.), Coll. Math. Soc. J. Bolyai 25 (1981).

[7] E HARARY, Graph Theory, Addison-Wesley Publ. Co. 1969.

[8] D. HETZEL, fSber regul~re graphische Darstellungen von aufl/~sbaren Gruppen, Diplomarbeit, Technische Universit~it Berlin 1976.

[9] W. IMRICH, Graphs with Transitive Abelian Automorphism Groups, Combina- torical Theory and its Applications II, Amsterdam 1970, 651-656.

[10] W. IMRICH, M.E. WATKINS, On Automorphism Groups of Cayley Graphs, Peri- odica Mathematica Hungarica Vol. 7(3--4) (1976), 243-258.

[11] L.A. NOWITZ, M.E. WATKINS, Graphical Regular Representation of Non-abelian Groups I,II, Can. J. Math. 24, Nr. 6 (1972), 993-1018.

[12] H. WI~LANDT, Finite Permutation Groups, London-New York 1964. [13] A.T. WHXTE, Graphs, Groups and Surfaces, New York 1973.

This paper was written while the author was visiting the University of Ham- burg at the invitation of Prof. HALIN.

Eingegangen am: 22.08.1990

Author's address: Ulrike Baumann, P~idagogische Hochschule 'K.F.W. Wander' Dres- den, Institut f'tir Mathematik und ihre Didaktik, Wigardstr. 17, PSF 365, 0-8060 Dres- den, Deutschland.

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Automorphisms of Cayley graphs