NEW ZEALAND JOURNAL OF MATHEMATICS Volume 24 (1995), 73-82

A CONSTRUCTION PRINCIPLE FOR SYMMETRIC GRAPHS

P e t e r Lo r i m e r (Received April 1993)

Abstract. Let G be a group acting symmetrically on a graph E. A method is de­ scribed by which a graph of the same valency as £ on which the direct product G r acts symmetrically can sometimes be constructed. In significant cases, particularly involving graphs of valency 3, further automorphisms outside G r can be found.

1. Introduction The context of this paper is the study of symmetry, particularly the symmetry of graphs. A group of symmetries of a graph can often be used to describe the features of more general structures. For example, the tessellation of 3-dimensional Euclidean space by permits its structure to be explained by one with its geometry and the Coxeter group [4, 3, 4] which is the group of symmetries of the graph of vertices and edges of the tessellation. The group of symmetries of any graph gives a precise measure of its symmetry. Among all graphs the most symmetric might be reckoned to be the vertex-transitive ones, those in which each pair of vertices are linked by at least one symmetry. Among the vertex-transitive graphs, measures of their relative symmetry might be taken from their vertex stabilizers, with more symmetric ones having bigger stabilizers, in some sense. Here, a way will be described by which the group of symmetries of a so-called symmetric graph can sometimes be used to construct a new one in which the stabilizer of each vertex is bigger. This is easy to do, provided the order and valency of the graph are not limited: in this construction the order increases but the valency does not. There are a number of notable cases in which this method works. For example: (I) The alternating groupA$ acts as a vertex transitive group of symmetries of the complete graphK$, of order 5 and valency 4, the stabilizer of each vertex beingA 4 . The construction leads to a group of order 7200 which acts as a group of symmetries of a graph of order 300 and valency 4, which turns out to be the graph of vertices and edges of a tessellation of 3-dimensional projective space by spherical dodecahedra: the stabilizer of each vertex is isomorphic toA 4 x C2 . (II) Vertex transitive graphs of valency 3 can be described in terms of 7 groups of automorphisms of the cubic tree,G 1, G\, G\, G3, G\, G5, where a subscript i denotes a group acting transitively on arcs of lengthi in the tree, [3]. The construction, beginning with one G\of or G2,i = 2 or 4, builds the groupG\ \ S2 or Gf I S2 which has Gi+1 as a subgroup. In this context, it is essentially the method used by Biggs in [1] to construct a of order 2352 from the of order 14 and, more generally, in [2]-

1991 AM S Mathematics Subject Classification: Primary 05C25, Secondary 20B25, 20F05. 7 4 PETER LORIMER

Notations and BackgroundLet G be a group of automorphisms of a graph E. Then (1) G acts vertex-transitively on E if, for all vertices u, v of E, G contains a member g with g(u) = v. (2) G acts symmetrically on E if for all vertices U\, U2 , Vi, V2 of E with u\ adjacent to u2 and v\ adjacent to V2 , G contains a member g with g(ui) = v\ and g(u2 ) = V2 . In the first case E is called a vertex-transitive graph and in the second it is called a symmetric graph. Suppose that G acts symmetrically on a graph E, finite or infinite. Let v be a fixed vertex of E, H the stabilizer of v in G and let a be a member of G which maps v onto an adjacent vertex. It is shown in [6], where more details can be found, that 1. a can be chosen so that a2 G H. 2. g(v) is adjacent to v if and only if g € HaH. I H\ 3. the valency of E is — — —- . J \H D H a\ 4. g(v) lies in the connected component of v if and only if g € (H, a); in particular E is connected if and only if G = (H, a). On the other hand, let G be a group, finite or infinite, H a proper finite subgroup of G, a € G — H and a2 € H. Then a graph E = E(G, H, a) can be defined as follows: the vertices of E are the left cosets of H in G and xH is adjacent to yH if and only if Hx~xyH = HaH. Each member g, of G, defines an automorphism of E by the rule g : xH —> gxH and, in this representation, G acts symmetrically on E. This does not mean that G is a group of automorphisms of the graph E. If N is the core of H in G, i.e. the intersection of all the conjugates of H in G, then every member of N fixes every vertex of E and it isG/N, rather than G, which should be' regarded as a group of automorphisms of E. The order of a group is the number of its members, the order of a graph is the number of its vertices. The number of edges at a vertex of a graph is its valency: if all vertices have the same valency, as is the case with vertex-transitive graphs, this number is also called the valency of the graph. In a group, ba = a~1ba, Ha = a-1 Ha. If G is a group and r is a natural number, then Gr is the direct product of r copies of G. |G| is the order of G. () denotes the group generated by whatever is between the brackets. If G is a group of automorphisms of a graph E and v is a vertex of E, then the stabilizer of v in G is the subgroup of G consisting of all g with g(v) = v. I am indebted to D.E. Taylor for conversations on the topics presented here, particularly for help on the structure of the graph of order 300 discussed in Section 3. A debt is also due to two referees who made this paper a better one than it would otherwise have been. A CONSTRUCTION PRINCIPLE FOR SYMMETRIC GRAPHS 75

2. The Graph Construction

In this section the construction will be described but a discussion of most of its properties will be left to Section 5. The two intervening sections contain details of the two applications mentioned in the introduction.

Let G be a group, H a proper subgroup of G, a e G — H, a2 e if, r a natural number and let Gr be the direct product of r copies of G.

Proposition 1.Suppose thatb = (az\ ,... ,azr) e Gr where z i , ... , zr lie in the centre of H(~)Ha and {z~1zj)a =(z~ 1Zj)~1 for i,j = l,...,r . If H is finite and K = {(/i,... ,h)-, he H}, then b e Gr - K , b 2 e K and

\K\ \H\ \K H K b\ \H D H a\ '

Proof. Clearly, b e Gr — K.

To show that b2 e K , consider (azi)2. First, (azi)2 = a2z?zi\ as z* e H fl Ha, also zfeHn Ha\ consequently, a2zfzi e H fl Ha. Thus, b2 e K if and only if, for all i and j, zfzi = ZjZj, a condition which is equivalent to the assumed condition (z~1zj)a = (z~1Zj)~1, as Z{,Zj commute.

To prove the last equality it will be shown that if k = (h,... , h) then k e KC\Kb if and only if h e H fl Ha. Suppose that k e K D K b. Then kb e K so that {haz\ . .. , haz') e K. Hence haZl = ... = haz- e H. Then, since e H ,h a e H. Hence h e H n H a. Conversely, suppose that h e H C\Ha. Then ha e HC\Ha and, since zi, ... ,zr lie in the centre of H fl Ha, hazi = ... = haZr e H. Thus kb e K and k e K C\ K b. |

From the groups G and Gr the two graphs £ i = E(G, H, a) and £ 2 = £(Gr ,K, b) can be constructed, using the method discussed in Section 1. By Proposition 1 they have the same valency but, if r > 1 , the order of £ 2 is greater than the order of Ei. The stabilizers of the vertices in the two graphs are the same, but in Section 5 some circumstances are described in which the order of the stabilizer of each vertex in the full automorphism group of £ 2 is larger than the stabilizer of each vertex of Ei in G.

While it is not essential for the development of this paper, it is worth contrasting this construction with two others: 7 6 PETER LORIMER

(1) In Gr put Hi = {(hi ,... , hr); hi,... ,hr e H }, ai = (a,... ,a) and S 3 = E(Gr , Hi,ai). Then the stabilizers of vertices in £ 3 have order \Hi\ = \H\r which is bigger than those of £ 1 , provided r > 1. But, balancing this, the valency of S3 is \Hi\/\Hi fli/f | = \H\r/\H D H a\r, which is bigger than the valency of S2 (if r > 1 ). (2) A degenerate case of Proposition 1 arises when each of zi, . . . , zr is the identity. Then, b — {a,... ,a) and it is clear that the isomorphism G —> Gr defined by 9 ~* (0,--., 9 ),

for each g 6 G, induces a graph isomorphism from S(G, H, a) to a connected component of S (Gr,K,b). As this latter graph is vertex transitive, all its connected components are isomorphic to S {G,a,H), the graph with which the construction began, and nothing new is obtained. An example which avoids this degeneracy is given in the next section.

3. Graphs of Valency 4 and Orders 5, 300, 18000 Here is a significant example of the way the construction works. In the alternating group, A$, let H be the stabilizer of 5 and put a = (23) (45). Then \H\ — 12, |H fl Ha\ = 3 and S(As, H, a) is a graph of order 5 and valency 4: it is the K$. For r = 2, let zi = 1, z2 — (123). Then, in the notation of Section 2, K — {(h,h)-, h e H} and b = ((23)(45), (13)(45)). As zi, z2 lie in the centre of H fl H a and (z^1 ^2 )“ = (zi lz2)~x, the graph S (A2, K, b) has valency 4 and order 602/12 = 300. As K and b generate A2, this graph is connected. The sphere S3, in 4- dimensional, space can be tessellated by 12 0 regular , forming a regular object called the 120-cell [ 6 ]. If S3 is mapped into 3-dimensional projective space by identifying antipodal points, the result is a tessellation of this space by 60 do.decahedra. The vertices and edges of this tessellation form the graph just constructed. D.E. Taylor and N.C. Wormald have found a Hamiltonian cycle in it (unpublished).

Next, taker — 3 and put zi = 1, z2 — (123), 23 = (132). These permutations form H fl Ha, which is abelian, and, as zf = z~l for i = 1,2,3, it is also true that {z~lzj)a = ( z ^ z j) - 1 for i, j = 1,2,3. Then b = ((23)(45), (13)(45), (12)(45)) and the graph E (A\,K,b) turns out to be a connected graph of order 18,000 and valency 4. In Section 5 it will be shown that its full automorphism group is bigger than A3. It is worth pointing out that this construction will not work with the full auto­ morphism group, S$, of the complete graph, K5, in place of A5 because, in S$, the role of H(~) Ha would be played by a subgroup isomorphic to the symmetric group, S 3 , which has trivial centre.

4. Symmetric Graphs of Valency 3 Following work of Tutte [7] it is now well known that there are just seven types of symmetric cubic graphs and they were described in [3] by the seven groups G1 , A CONSTRUCTION PRINCIPLE FOR SYMMETRIC GRAPHS 7 7

G\, G\, G$, G\, G 4, G$ each generated by three elements h, a, p which satisfy

h3 = 1 , p2 = 1, and a2 = l for Gu GlG3,GlG5, or a2 = p for G2,G2. and

G1 : p = 1, Gl and G\ : = P, hp = h r1',

G3 : ph =P, pa = q, hq = h~x qp = pq\ G\ and G\ : pa = p , ph = q, qa — r, hr = h~\ qh = pq, (p, (?) and (p, r) are elementary abelian groups, rq = pqr; G5 : ph =p, pa = q, qh r,— r a = s, hs = h ~ \ rh = pqr. (p, <7, r) and (p, <7, s) are elementary abelian groups, sr = pgr s.

In these groups, a subgroup H is defined by the following table, which also describes H D H a.

Gi G IG 2 G3 G IG 2 g 5

H (h) {h,p) {K p ,q) (h,p,q,r> (h,p,q,r, s) H H H a 1 (V) (p,q) (p,q,r) (P, q,r , s)

If G is any one of these seven groups, the graph E(G, H, a) is the infinite cubic tree. Their significance is that they are the only symmetric groups of automorphisms of the infinite cubic tree in which the stabilizer of a vertex is finite: for a group Gs, G] or G2 the order of H is 3 x 2s 1. Further details can be found in [3]. Djokovic and Miller [5], show that, as groups of automorphisms of the infinite cubic tree, G\ and G\ are subgroups of Gs while G\ and G2 are subgroups of G5 , in all cases the index being 2 . Let G be one of G\, G2, G\ , G2 and put z\ = 1, zq = p. In terms of Section 2, b = (az\,az 2 ) = (a, ap). As p lies in the centre of H D Ha and pa = p — p-1 , the construction in Section 2 works to give a graph £(G 2, K, b) where K = {(h, h)] h € H}. As it is clear that the connected components of X^G2, K, b) must also be infinite trees, the purpose of this example is not the construction of new graphs. 7 8 PETER LORIMER

Rather, it is in the description of further automorphisms of this tree induced by the automorphism a of G2 given by

o' : {91, 92) -> {92,9i)-

This is described in general terms in Section 4 and this example is discussed again in Section 7. 5. The Stabilizer of a Vertex The main point of the construction of Section 2 is that, in some circumstances, the stabilizer of each vertex in the new graph can be bigger than that of the old one. In that the vertex stabilizer of a vertex transitive graph provides a. measure of its symmetry, this suggests that the new one has more symmetry than the old. This section contains generalities which are illustrated in the remaining two sections. The construction here is given in terms of the wreath product G\X of a subgroup X of a symmetric group Sr on 1,... , r and a group G. This group G I X consists of all products of the form {9l,••• , 9r)& where (<7 1,... ,gr) G Gr and a G X. Multiplication is determined by the equation

& {9 l, • • • , 9r) — {9cr( 1) > • • • j 9cr(r) )&•

As in Section 2, let G be a group, H a proper subgroup of G, a e G — H and r a natural number. Let z 1 ,... , zr be members of the centre of H (1 Ha with the property that {z^lzj)a = {z~lzj)~l for z, j = 1, ..., r. Let b = (az\ , ... , azr) and K = {{h,... ,h);heH}. If X is a subgroup of Sr, then KX is a subgroup of G I X, b € G I X — KX: and b2 G KX hence a graph T,{G I X,KX,b) can be formed. The next Theorem describes a condition that E (Gr,K,b) and Y,{G I X,KX,b) are the same graph. With this notation, Gr and X both appear as subgroups of Gl X and Gr P\X = 1 . Theorem 1.Suppose that

Z i Z a(x) = zj Z cr(j) for i, j — 1, ..., r and for each a G X . Define a function ip from the vertices of E(Gr , K,b) to the vertices of E(G \ X, KX,b) by the equation

if>{gK) = gKX.

Then ip is a graph isomorphism. Proof. It is clear that ip is a one-to-one mapping between the vertices of the two graphs. Suppose that g\K and g2K are adjacent in E {Gr,K,b), i.e. K g ^ g iK = KbK. Then X K g ^ g iK X = X K bK X and giK X , g2K X are adjacent in E (GlX, K X , b). A CONSTRUCTION PRINCIPLE FOR SYMMETRIC GRAPHS 7 9

To prove that is an isomorphism, it is sufficient to prove that the two graphs have the same valency, i.e.

I * * | _ 1 * 1 \KXn(KX)b\ \KHKb\'

To this end, suppose that a E X. Then

cr6(7_ 1 = (zf 1 a _1, ... , z~1a~1)cr(azi,... , azr )cr- 1

= (zf 1 a*~1, ... , z~1a~1)(az(T(i),... ,az<7(r))<7(T- 1

= (-^1 %

From the assumption in the statement of this theorem,

zl za{ 1) — • • • — Zr Za(j.y

Thus ab(j~l E K and, as z\ , ... , zr E H fl H a, also aba~x E K ft K b, i.e. ab E

(.K D K b)a . Suppose that g € KX fl (KX )b, say g = k\a\ = w^h k\, k2 € K and ui ,

1 * * 1 _ 1 * * 1 \KXn{KX)b\ \{KnKb)X\ \K\\X\ \K n K bn X \ ~ \Knx\ \KnKb\\x\

1 * 1 \K n K b\'

Thus, the valencies of the two graphs are equal. I This theorem represents the graph E(Gr, K, b) as E(G IX, K X , b) and the stabi­ lizer KX is bigger than K provided only that X ^ 1. However, the construction of a graph Y,(G,H,a) does not necessarily represent G faithfully as a group of auto­ morphisms of E(G, H, a): the representation will have kernel the intersection of all the conjugates of H in G, a normal subgroup of G called the core of H in G. The next Theorem shows that, under the hypotheses of Theorem 1, the core of KX in GI X is the same as the core of K in Gr. Theorem 2. Under the hypotheses of Theorem I, if r > 1 then the core of K in Gr and the core of K X in G I X are both equal to the subgroup of Gr consisting of all (h,... ,h) with h in the intersection of H and the centre of G. Proof. Denote the core of K in Gr by N\ and the core of KX m GIX by N2. Suppose that k = (h,... ,h) E Ni. If g — (gi,... , gr ) E G then k9 E N\ C K i.e. (h9lr .. . ,h9r) E K. 80 PETER LORIMER

Hence, h91 = ... = h9r. As this is true for arbitrary gi ,... , gr € G, it follows that h lies in the centre of G. As this argument is reversible, the result is proved for N\. The argument in the previous paragraph also shows that N2 C\ Gr = Ni. Suppose that a € N2 fl X, a ^ 1, and that g = (<7 1,... ,gr) € Gr. Then a9 € N2 so that a 9 — k<7 \ for some k € Ni, say k = (h, ... ,h) and some

($1 9 a( 1 ) i 9 r 9 (7^ ) } (^, ••• 1 f y i C — (T1 .

Thus, = gih, i = 1,... , r. As c ^ 1, there is a number i with a(i) ^ i. Let hi be any member of H and let u be any member of G — H. Choose a member g = (gi, ... , gr) of Gr with gi = hi and ga^ = u. Then u = hih for some h € H. Thus u € H, a contradiction. Thus N2 fl X = 1. Hence N2 = N i, which proves the Theorem. I The specialization of this result to the significant case that G is actually a group of automorphisms of E(G, H, a) is contained in the next theorem. Theorem 3. Under the hypotheses of Theorem 2, ifG is a group of automorphisms of £, then G I X acts faithfully as a group of automorphisms £(G ofI X , K X , b).

Proof. If G is a group of automorphisms of £, then the core of H in G is 1 . Thus, the intersection of H and the centre of G is also 1 and Theorem 2 implies that the core of KX in G I X is also 1. Consequently, G I X acts faithfully on E(GlX,KX,b). I

6 . The Graph of Order 18,000 Section 4 will now be illustrated by the graph E = K, b) constructed in Section 3. In discussing subgroups of the wreath product A5 I S 3 it will avoid confusion if S 3 is regarded as acting on the set {a,/3,7 }. Then b = (aza,azp, azy) = ((23)(45),(13)(45),(12)(45)). Let X = {1 ,(a/?7), ( 0 7 /*)}. For cr = (a/3 7 ), za lza(a) = zcr(/3) = z^ ^(-y) = (123) and for

zi za(i) = z2 z

il>{h) = {hq,W) tp(p) = a V>(Q) =°ip,p) ip{r) = a{q,q) xp(s) = a(pr,pr) ip(a) = a(a, ap)

define an isomorphism from G$ onto a subgroup of G1S2, (where G = G\), under which the action of G$ on £(Gs, H,a) is compatible with the action of GI S2 on X (G lS 2,K S 2,b).

Proof. To avoid confusion between the generators of G 4 and G5, the standard gen­ erators of G5 introduced in Section 2 will be written as ho,ao,po,qo,ro,so instead of h, a,p, q, cr, s. In the group G I S 2 , put hi = (hq,hq), ai = cr(a,ap), pi = cr,qi = cr(p,p), 7*1 = cr(q,q), s\ = cr(pr,pr). Then it is easy to show that all the relations for G5 are satisfied with hi, a\,p\, q\, ri, si in place of ho,ao,po,qo,ro,so, respectively. Thus, there is a unique homomorphism, from G5 into G I S2 mapping each of these generators of G 5 into the corresponding element of GI So­ under the action of the image of

H0 = (po,qo,ro,so,ho) is (o,

^(£5) = {(p,p),{q,q),(r,r),{h,h),(a,ap))S2 = M S 2 where M = ((p,p),(q,q),(r,r),(h,h),(a,ap)) 82 PETER LORIMER

As all the relations for G\ are satisfied with ap in place of a, the mapping of each member of M onto its first co-ordinate produces an isomorphism r : M —► G\. Thus, M is a subgroup of index 2 in MS 2 which is isomorphic to G\. Now, ^(Pitfi) = (p,p), ip(piri) = (q,q), ^(qiSi) = (r,r), ^(/ifri) = (/i,/i), piai) = (a, ap). It is straightforward to show that if pi

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Peter Lorimer University of Auckland Auckland NEW ZEALAND [email protected]