CAYLEY GRAPH ENUMERATION

Marni Mishna

BMath, University of Waterloo, 1 998.

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March 2000

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Abstract

Pol y a's Enurneration Theorem is a powerfùl method for counting distinct arrangements of objects. J. Turner noticed that circulant graphs have a sufficiently algebraic structure

that Polya's theorem can be used to determine the number of non-isomorphic circulants of order p for prime p. Recent r d t s on CI-groups suggest that Turner's metbod can be used to enurnerate a larger collection of circulants, circulant digraphs, and Cayley graphs and digraphs on Zg and Z*

Acknowledgments

Brian Alspach found me a nice enumeration problem and carenil1 y read the work. NSERC and SFU provided me with the financial support that allowed this to be a quick project.

Big huge thanks to the wonderful, fun people that 1 have met whïie in Vancouver. 1 have

been enlightened, enteitained and inspïred. I will leave here a better person than when 1

arrived.

Karen Meagher and Adam Fraser, my best Wends, have continued to tolerate, support and encourage me. Karen offered some real killer suggestions and actually read the whole thing.

This one goes out to my family, especially my mom who even tried to understand what it

meant.

it 's an automatic toaster! b m s delicious coffee automatically! does any mlxing job!

by solving cornplex muthematical fonnulas

Contents

Approval

Abstract

Acknowledgments

Contents

List of Tables

List of Figures

ii

iii

1 Introduction 1

1 . 1 Definitions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 CI-Groups.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Some Known CI-Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Circulants of Prime Order 5

2.1 Determinhg the Cycle index . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Enumerating Circulants of Prime Order . . . . . . . . . . . . . . . . . . . 1 2

. . . . . . . . . . . . . . . . . . . . . . 2.3 Circulant Digraphs of Prime Order 14

. . . . . . . . . . . . . . . . . . . . . . . 2.4 Counting Regular Cayley Graphs 15

3 Circulants and Circulant Digrapbs 18

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Circulants 19

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Circulant Digraphs 26

4 UnitCirculants 28

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Odd Prime Powers 29

. . . . . . . . . . . . . . . . . . . . . . . . 4.2 Products of Odd Prime Powers 31

. . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Unit Circulants of Al1 Orders 33

5 Cayley GraphsoverZ. x Zp withp prime . 37

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Rational Normal Form 38

. . . . . . . . . . . . . . . . . . . . . . . . 5.2 'Ihe Size of a Conjugacy Class 43

. . . . . . . . . . . . . . . . . . . . . . . . . 5.3 TheCycleIndexofGL(2. p ) 44

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Z:andBeyond 52

. . . . . . . . . . . . . . . . . . . . . . . . . 5.5 TheCycleIndexofGL(n. p) 52

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 FuialThoughts 54

Bibliography 55

List of Tables

5.1 Sizes of Conjugacy Classes in GL(2 , p ) . . . . . . . . . . . . . . . . . . . 44

vii

List of Figures

viii

Chapter 1

Introduction

1.1 Definitions and Notation

Determining the number of distinct graphs in a given family is one of the most basic ques-

tions one can ask about a family of graphs. Graph theorists have devoted much energy

to searching for elegant answers to the graph isomorphism problem for many families

of graphs. Polya's theorem of enurneration, when it first became widely appreciated in

the eariy 196Os, serveci as the main tool for many graph isomorpbism problems. In 1967

I. Turner determined that a class of Cayley graphs was well suited to this approach. Cayley

graphs are defined in relation to groups and consequently have a usefbl underlying structure.

The Cayley graphs Turner considered possess a particular property, they are Cayley graphs

on CI-groups, and recent work in this area has found more families with this characteristic,

thereby opening up the possibility of applying his methods to these new families.

Throughout, 9 shall denote the Euler phi-function. Hence @(n) is defined over the

natural numbers as the number of integers i, 1 5 i 5 n, coprime to n. The additive cyclic

group of order n will be denoted by Zn, and Zn will always denote the multiplicative group

of units of the ring of integers modulo n. For a group G let Aut(G) denote the group of

automorphisrns of G. A graph automorphism is an adjacency preserving permutation of the

vertex set. We will use similar notation Aut (X) to denote the graphs automorpbisms of the

Figure 1.1 : CAYLEY GRAPH X(&, {r, r3, r2 t ) ) AND CAYLEY DIGRAPH X ( & , { r , r2 t ) )

graph X.

The next two definitions describe a Cayley graph.

DEFINITION. A Caylqsubset S o f a group G is aninverse closed subset (s E S s-' E S) of G not containing the identity.

DEFINITION. A Cayley Graph is represented by X(G; S) where G is a group, and S is a Cayley subset of G, also known a s the connection set. The Cayley graph has vertex set G and edge set

{(91,92)191 = a s , s E S).

DEFINITION. A Cayley digmph X(G; S ) is defined on a group G and a set S C G \ e. It

has vertex set G and there is a directeci edge fiom gl to g2 if and only if gz = g1s for some s E S.

EXAMPLE. The dihedral group Ds = { r , t Ir4 = t2 = e, tr = r-' t) is a fine p o p upon

which to define a Cayley graph and a Cayley digraph. Figure I .1 provides an example of a Cayley graph and a Cayley digraph on D g .

Our ultimate goai is to provide some enurneration results for some familia of Cayley

graphs. This is most easily accomplished when there is a useful relationship between the

isomorphisms of the graphs and automorphisrns of the group.

If a E Aut(G), then for S C G, let a(S) = {a(s) 1s E S). A group automorphism a

of G can also be newed as a map between two Cayley graphs on G upon considering the

resulting action of a on the vertices G. That is, there exists a map between X(G; S) and X(G; <r(S)). In fact, this action cm be a graph isomoqhism.

1.1 THEOREM. Let A' = X(G; S ) bea Cayleygruph on thegroup G. I f a E Aut(G). then

a is an isomorphism@m the graph X to the gmph X' = X(G; a(S))

PROOF: By the definîtion of Cayley graph we have that UV E E ( X ) if and only if v = us

for some s E S. Since a is a group automorphism, we have that v = us if and only if

a ( v ) = a(u )a (s ) . Thus vu E E ( X ) if and only if a(v)a(u) E E(X') so a is a graph

isomorphism between the graphs X and X'.

Next we consider the reverse relationship. 'ïhat is, in which cases does an isomorphinn

of the graph correspond to an automorphism of the group?

DEFINITION. Let G be a finite group, and let X = X ( G ; S) and X' = X ( G ; s') be Cayley

graphs on G. S satifies the Cayley isomophism pmperty if whenever X is isomorphic to

X', there exists a group automorphism a of G such that a is also a graph isomorphism from X to Y. We abbreviate this as S is a C h b s e t .

DEFINITION. If every Cayley subset of G is a CI-subset, then we Say that G satisfies the

Cayley Zsomorphism Pmperty, or succinctly, G is a CI-gmup.

This definition cornes from a generalization of properties of circulant graphs. The rela- tionship and motivation wiI1 soon become clear.

1.3 Some Known CI-Groups

The Cayley Isomorphism property is i n d d a nice property, almost suspiciously so. It is in

our best interests to determine which groups are CI-groups, indeed if any exist at dl, as this will facilitate enurneration greatly.

The search for existence best begins with the cyclic groups. J. Turner [ 161 began the search with cyclic groups, and obtained results for cyclic groups of prime order. M. Muzy-

chuk [9] settled the case for the cyclic groups in general.

1.2 THEOREM (Muzychuk). The cyclic p u p s which are CLgmups are preciseiy rhose of

order n where n is 8,9,18 or n = 2% where e E { O , 1,2} und m is odd and squoref;eee

The next obvious set to consider is the products of cyclic groups. We have the following

results in this w e . C. Godsil [4] managed a partial answer to the product of two cyclic

groups.

1 3 THEOREM (Godsil). The p u p Z;, p p h e . is a Ci-group.

The following result was determined independentiy by both T. Dobson [2] and M.-Y. Xu [I7].

1.4 THEOREM (Dobson, Xu). The group ZE, ppnme b a C I - p u p .

It is these CI-groups that we shall consider in this thesis. We shall determine the number of isomorphism classes of Cayley graphs and Cayley digraphs of order n on these groups:

Zn with n as in Theorem 1.2, Z: and Z: for p prime.

Recently [8], it has been shown by J. Moms and T. Dobson that Z; is also a CI-Group.

The reader, at this point, may hypothesize that Z," is a CI-GToup for al1 n. In fact, L. Nowitz

showed [IO] that Z I is not a CI-Group. However, for al1 n such that it is true, the methods presented in this work will determine the number of isomorphism classa of a given order.

Chapter 2

Circulants of Prime Order

Circulants are an interesting class of Cayley graphs well worth studying They are Cayley graphs on the simplest of groups and thus may provide direction and insight into Cayley graphs on other groups, particularly finite groups. Al1 h i t e vertex-transitive graphs of prime order are cimilants, hence the study of vertex transitive graphs can gain fkom the study of circdants.

DEFINITION. A cimr[ant is a Cayley graph on a cyclic group. We denote the circulant

X(Z,; S) by simply X ( n ; S).

The complete set of circulants of order 5 is illustrated in Figure 2. Notice that X(5; {2,3))

and X (5; { 1,4)) are isomorphic.

We shall begin with a known result about the number of circulants up to isomorphism

Figure 2.1 : THE COMPLETE SET OF C~RCULANTS OF ORDER 5

of prime order p. A very elegant relationship, that of Theorem 1.2, exists between graph

isomorphisms and group isomorphisms which allows us to use strong group theoretic twls.

Further, these rnethods, notably PSlya's enumeration theorem, can be used to enurnerate circulant graphs of other orders.

Enumerating the prime case directly offers sufficient insight that we include it. First we show that the prime circulants are the complete set of vertex-transitive graphs of prime

order.

2.1 LEMMA. Cayley graphs are vertex-transitive.

PROOF: The action of a group G acting on itself by left multiplication is a permutation.

This action clearly presewes adjacency in any Cayley graph defined on G. The permutation

which takes u to v is left multiplication by vu-' E G.

Defme the mapping TaT6 acting on Z, to be Taeb(x) = ax + b. I f a E Z ,, TaVb is a permutation

of Zn. The notation H < G indicates that H is a proper subgroup of G.

2.2 THEOREM (Burnside). IfG is a transitive pennutation group acting on a prime number

p ofpoints, then either G is doubly transitive or

2 3 COROLLARY. I f G is a îmnsitive permutation group acting on a prime number p of

points and G is not doubly transitive, then G contains a unique slrbgmup oforder p.

PROOF: First we establish existence. The subgroup generated by (Tl = {TIS61b E Z p ) is of order p and containeci in G by Burnside's Theorern.

Consider TaBb(2) = ut + b. Notice that if a # 1, (a - 1) is a unit and has an inverse in

which case ( a - 1)-' ( p - b) is a fixed point for Tava (x ) - Since G is acting on p elemmts and

CHAPTER 2. ClRCULANTS OF PRIME ORDER 7

the order of a permutation is the Least common multiple of the cycle lengths in its disjoint

cycle decomposition, an element of order p in G must be a p-cycle. Hence, an element of

order p has no fixed points and the result follows fkorn Theorem 2.2.

2.4 THEOREM (Turner). A graph X of order p is vertex-transitive if and oniy if X is a

c i d a n t graph.

PR00~:[16] Since a circulant is a Cayley graph one direction folîows fkom Lemma 2.1.

Now suppose that X is a vertex-transitive graph of prime order p. If the automorphism

group is doubly transitive, X is either the complete graph or the empty graph, both cir-

culant~. Otherwise, we may assume that Aut(X) is not doubly transitive, and thus has a unique subgroup of order p. We may consider a labelling of the vertices such that the gen-

erator of the this unique subgroup is p = (O I . . . p - 1). If i is adjacent to j in X, then they must be adjacent under the image of automorphimi p, thus, i + 1 is adjacent to j + 1,

and upon considering powers of p, i + s is adjacent to j + s for O 5 s 5 p - 1, with al1 computations modulo p. If we let S denote the set of vertices adjacent to the vertex labeiied

O, we have i is adjacent to j if and oniy if i - j and j - i are adjacent to O if and only if

i - j E S and j - i E S. Clearly X is a circulant of order p with connection set S.

HaWig illustrated a motivation for studying circulants, we now r e m to the idea of

enurneration. The next result is a specification of Muychuk's Theorern to the prime case.

2.5 THEOREM (Turner). Two circulant paphs X = X ( G ; S), and X' = X(G; S') of

prime order p are Lromorphic ifond only iftherir exists some a E ZP such that as = S'.

PROOF: First suppose that there exists some a E Z; such that S' = as. Lefi multiplication

by a is a graph isomorphism fiom X to X'.

Now suppose that X and X' are isomorphe as graphs with isomorphism f : V ( X ) + Vp(X'). If X is either the completegraph Kp or its complement Kp, then S = {1,2,. . . , p -

1 ) or { ), respectively. in either of these cases S' = S. So, consider the case that X is

neither Kp nor K p . For any circulant on vertices 0,1, . . . , p - 1 the rotation permutation

p = (O 1 . . . p - 1) is a graph automorphism. The subgroup generated by p has order p.

Now, p is an automorphism of both X and X'. Funber, since f is an isomorphism,

a = (f (O) f (1) . . - f ( p - 1)) is also an automorphism of X' generating a subgroup of

order p.

Since X is neither Kp nor &, its automorphism group is not doubly transitive. Hence, by Corollary 2.3 we have that the group of permutations of X' contains a unique subgroup

of order p. Thus, a E ( p ) and we have that

Thus, if we rewrite o we can see that for some O, 5 i 5 p - 1, f (i) = O, f (i + 1) = a and in generai f (i + j) = a j. We can now prove the theorern upon examining the adjacency

criterion for circulants.

s E S i is adjacent to i + s in X for al1 i E Z,

f (i) is adjacent to f (i + s) in X'

o a(k + i) is adjacent to a(k + i + s) in X' o a(k+i)-a(k+i+s) E S'

as E SI

CHAPTER 2. CIRCULANTS OF PRIME ORDER

2.1 Determining the Cycle Index

Theorem 2.5 provides the necessary information about the structure of circulants to appeal

to Polya's enumeration theorem.

In a fiamework suited to counting with the Pdya enumeration theorern, there is a do-

main for which one has an understanding of a permutation group, and a range of acceptable

values. Polya's theorem states that we c m detemine a generating function for the num-

ber of distinct functions from domain to the range, using the automorphism group and the

generating function of the range.

Let the domain be set D and let the range be set R. Define RD to be al1 of the functions

fiom D to R. The theorem counts the number of distinct functions fiom D to R. A per-

mutation group G acting on D will induce a permutation action on R! Polya's theorem

reduces counting the orbits of the induced action on R ~ , that is, the number of distinct

functions, to counting the distinct actions of G on D. The main tool is the cycle index of

G, which holds the information of the permutations of D and which we define next.

Let G be a permutation group acting on a set R. Consider the disjoint cycle decomposi-

tion of o E G. Suppose it contains precisely bk cycles of length k, for 1 5 k 5 151 1. We de-

fine a monomial n ( ~ ) associated with the decomposition as follows: n ( ~ ) = zb,lz? . . . x m ,

wherem = 101. Note that bl +2b + ...+ kbk = m.

DEFINITION. The cycle index Z(G, R) of the permutation group G acting on R is defmed

to be the polynomial in indeterminates x I , xz, . . . , xm

EXAMPLE. Let G be the permutation group generated by the 6-cycle (O 1 2 3 4 5).

The monomial corresponding to the identity permutation is r(Ic) = x:. The monomial

corresponding to a 6-cycle, of which there are two in G, is xi. The two permutations which

are products of two disjoint cycles of length three correspond to the monomial x;. The

remaining permutation is a product of three disjoint 2-cycles. The cycle index for G is thus

1 6 Z(G, {O? 1,2,3,1,5)) = 6 ( ~ i + xi + 2': + 2x5).

The next example detemiines the cycle index of the group of automorphisms of a cyclic

group acting on the cyclic group.

EXAMPLE. Let p be prime and let Z , the multiplicative group of units of Z, act on the set Z, \ O by multiplication. Theorem 2.6, illustrates that a E Z; generates a group of

permutations of the elements of Z,. We can determine the cycle index of this action.

DEFINITION. Let a , n E Z with gcd(a , n ) = 1 . We definite the onfer of a mod n to be

the least positive integer k ( a ) such that a'(") = 1 ( m o d n ) and we denote this by ord,(a).

Notice that this is equivalent to saying that a has multiplicative order k(a) in the group Zn

2.6 THEOREM. Let G be a cyelic p u p oforder n. niegrvup Aut(G) of automorphisms of

G is exactIy the p u p ofaii au tom or ph^ { a k : a i ( g ) = gk , 1 5 k < n, gcd(k, n ) = 1).

Moreovec the mapping k I+ a k is un isornorphism f i p m Z:, to Aut (G) .

PROOF: [13] First we ver@ that a k E Aut (G). We can see that a k is a homomorphisrn of

G, since for g l , g2 E G, we have ak(g1g.l) = ( g i 9 2 ) k = gfg$ = a k ( g l ) a ~ ( g l ) . Let z be

a generator of the group and let g, = x and g2 = x', O 5 s, t < n. Now suppose that

a k ( g 1 ) = a k ( g Z ) . Then we have that xtk e zsk ( m o d n ) . Thus, t k = sk ( m o d n ) . Since k and n are coprime, t i s (mod n). Given the possible values for s and t, they m u t be

equal, and hence gl = g2 and a c is injective. Given z t , clearly ak (zk-lt ) = x t , implying that a k is surjective.

Next we veri@ that any element of Aut(G) is a k for some k , gcd(k , n) = 1. Let x be a generator of G and let a E Aut (G) . Since a(zt) = ( a ( z ) ) ' , the action of a is completely

detexmineci by its action on X. As a ( x ) must generate G as well, it has order n. Thus, a ( x ) = xk for some k with gcd(k, n) = 1, as these are the generators. From this we sce

that for any element zt of G, a(zt) = ~ ( x ) ' = (xk ) ' = a k ( x t ) .

2.7 LEMMA. The number of elemena oforder k in Zn b @ ( k ) ifkln andzerv otherwise.

PROOF: Certainly the order of an element must divide the order of the group and so we

consider only those k which divide n. The elements of order n are precisely the integers in

Z CO-prime to n [ 1 31. Thus, there are O (n ) elements in the cyclic group of order n which

have order n. If a E Z n is of order k, it generates a subgroup of Zn of order k. From our

remarks about n, there are at least O(&) elements of order k.

Now, Z n is a cyclic group of order n, hence there exist subgroups of order k for each

divisor k of n. Since for any positive integer n, CkIn 9 ( k ) = n, we can account for wery

element in Zn upon considering the subgroup they generate. Hence, the number of elements

of order k in Zn, for k a divisor of n, is 9 ( k ) .

2.8 THEOREM (Gauss, [ 141). The gmup of units of Z, is a cyclic gmup. isomorphic to the

cyclic p u p of order p - 1.

2.9 THEOREM. î l e cycle index of Z, acting on Zp \ O is

where the sum is taken over all divisors of p - 1.

PROOF: Let the order of a modulo p be k ( a ) . We know that for a 2 1, k ( a ) = ( p - l ) / g c d ( a , p- 1 ) . If x is any element of Z,\O, then it is in the cycle ( x a z a 2 z . . . a'(")-lx).

The permutation g.(x) = ax splits Z ; into cycles each of length k ( a ) . Since k(a) must

divide p - 1, there are (p - 1 )/ k ( a ) cycles of length k (a ) .

Summùig over the elements of Z;, we have the cycle index

2.2 Enumerating Circulants of Prime Order

Polya's Theorem brings toge& the three necessary cornponents for counting in a very

simple way. To establish context, fvst recall the Cauchy Frobenius Lernma ( a h known by

the Burnside Lemma). If G is a finite group of transformations acting on a finite set R of

objects, and two objects are considered equivalent if one is transfonned into the other by

a transformation in G, then the number of inequivalent objects is & COEC f iz(g), where

/ix(g) is the number of points in R fixed by o.

Polya's Theorern generaiizes this idea. Let D and R be finite sets. Let RD denote the

set of al1 h c t i o n s fiom D to R, and let G be a permutation group acting on D. For each

O E G define a permutation a acting on RD by

for ail f E R~ and al1 z E D. The mapping taking O to a is a homomorphism of G to a

permutation group G.

2.1 0 THEOREM (Pdlya). Wth D, R, G, G. and R~ asper the above discussion, the number

of inequivalent f E RD under G is detennined by evaluating the cycle index of G with -ch variable set to 1 RI.

The theorem is acnially much stronger. A good discussion exists in either [ 1 ] or [ 121.

To ssummarïze, if the weight of an elernent y in R is w(y), then the generathg function for

R is then g ( u ) = C,,, uw(y). The generating fiuiction counting orbits of f E RD unda G is ob tained by substituting g(u') for xi in the cycle index of G. In pariicular, the number

of orbits of the permutation group G is given by setting al1 of the weights equal to one,

essentially, evaluating the cycle index of G with each variable set to 1 RI.

We can apply Polya's Theorem to count the number of circulant graphs of order p, p an odd prime. Let the domain D be the set of ordered pairs

with the range R chosen to be the set (0,l) . A fbnction fs in RD corresponds to a circulant

graph X ( n ; S) in the following way. The set {i, -i) is in S if and only if fs({i, 4)) = 1.

The permutation group acting on D that we will consider is Z; with the action of multipli-

cation.

Since multiplication by a is the same as multiplication by -a with respect to D, in this

case the permutation group can be pared down to Z,- / (1, - 1) i Z e. 2

2.1 1 THEOREM (Turner). Let p be a prime. The number of circulant graphs of order p, to

within isornmphism, is CI

PROOF: We have by Theorem 2.5 that two circulant graphs X(Z,; S), and X'(Z,; T) are

isomorphic if and only if their comection sets satisfy S = UT for some a E 2,'. Con-

sider the action of a in the cyclic group of order (p - 1)/2 on D. This maps the set

{i , -i ) to {ia, - ia) . This induces an action ?ï on RD defined by a( f ) (z) = f (as). No-

tice that f({ai, -ai)) = 1 if and only if a(f)({i, -i)) = l for each i € 2,'. Since

a( f) E RD, a( fs) = fi for some T C D, with S = aT. Thus, counting the number of

orbits of this group simultaneously counts the number of isomorphian classes. The cyclic

index of the latter group is

CHAPTER 2. CIRCULANTS OF PRIME ORDER 14

We now substitute 2 = 1 Rl for each variable, according to Polya's Theorem, and obtain the

desired result.

EXAMPLE. TO illustrate this result consider the circulants on five vertices. There is the

empty graph, the five-cycle and the complete graph; the choices of valency 0, 2 and 4,

respectiveiy. If we now wnsider the number of classes according to the formula, we have

as is consistent with Figure 2.

EXAMPLE. TO illustrate the power of the formula we consider a more impressive result.

Let p = 53. The cycle index for this case is

and it follows that the number of non-isomorphic circulant graphs of order 53 is

2.3 Circulant Digraphs of Prime Order

The enumeration of digraphs is simpler than the enumeration of graphs, yet has not pre-

viously been investigated. Theorem 2.5 holds in the digraph case as well, and hence we

can use the same methoâ. However, in this situation the comection sets can be any subset

of Z, \ O, so we need not concexn ourselves with dealing with pairs (i, - 2 ) . Instead we

simply determine the cycle index of ZP acting on Z, by lefi multiplication as calculated in Theorem 2.9. When the order of a circulant is prime there is not much difference between

the directed and the undirecteci case. However, the r d t s for digraphs are typically simpler

than the results for graphs.

2.12 THEORE M. The number of non-isomorphic cimlant digmphr of order p. p prime. is

EXAMPLE. We can contrast the earlier result with the number of circulant digraphs on five

vertices. The number of non-isomorphic digraphs on 5 vertices is

EXAMPLE. It is just as simple to detennine the same information of a circulant digraph of

a larger order. Consider p = 29. The number of non-isomorphic circulant digraphs of order

2.4 Counting Regular Cayley Graphs

One of the charms of this counting method, as opposed to, say, directly appealing to the

Cauchy-Frobenius Lemma, is that we can obtain even more relevant information. In fact,

we have not made use of the true power of the more general version of P61ya7s Theorem at d l . The results so f a . could just as easily have been wmputed with the Cauchy-Frobenius

Lemma.

Instead of producing the number of orbits, which in this case yields the number of non-

isomorphic graphs, we can greate a generating function which gives us more information

about these graphs. For example, since Cayley graphs are vertex transitive they are regular, that is every vertex has the same valency. We cm use Pdya's Theorem to create a generating

function F ( x ) where the coefficient of uk is the number of non-isomorphic graphs which

are k -regular. A Cayley graph is k -regular when the connection set is of size k.

To get the desired generating function from Polya's theoran we need a suitable gener- ating function for R = {O, 1). Recall that fs(a) = 1 if and only if {a, -a) C S. Since the

C W T E R 2. CLRCULANTS OF PRIME ORDER 16

valency of a graph is the size of its comection set S, and for each choice of 1 for a function

the size of S is incremented by 2, the element 1 in R should have a weight of 2, and O a weight of O. Thus, the generating fwiction for R is G(u) = 1 + u2. According to Polya's

theorem if we substihtte G(uk) for x k in the cycle index, the result will be a generating

function for the number of k - regular graphs of a aven valency.

EXAMPLE. In the case of circulants of order 5, we had the cycle index of $(x: + x2). Thus,

upon replacing each xi with 1 +u2' we have the generating function i((l + ~ ~ ) ~ + ( l +u4)) =

1 + u2 + u4. This suggests a circulant each of valency 0,2, and 4, which was also evident

fiom Figure 2.

If two Cayley graphs are isomorphic their complements are isomorphic under the same isomorphism. Thus, for a given group G, if there are m non-isomorphic, k-regular, Cayiey

graphs on G of order n, there are also m non-isomorphic Cayley graphs on G of order n

of valency n - k. Thus one oniy needs half the terms in the generating fùnction to get the

complete picture.

EXAMPLE. In an earlier example we calculated that there were over two million circulants

of order 53. The vaiency generating function,

indicates how they are distributed. For example, there are 25 300 14-regular circulants of

order 53.

We c m even detennine this formula in general. If we denote the coefficient of un in a polynomial f (u) by [un] f (u ) , then the numba of circulants of order p and vaiency k is

For example, the number of circulants of prime order p of valency two is always 1. The

CHAPTER 2. CIRCULANTS OF PRIME ORDER

number of valency 4, is (p - 1)/1 if four divides p - 1 and (p - 3)/4 otherwise.

We can use this same technique in the w e of digraphs to count the p p h s of a given out-valency.

The number of circulant digraphs of order p and out-valency n is

Chapter 3

Circulants and Circulant Digraphs

The case of countïng circulants of prime order is so pleasingly solved it would be satis* ing if the composite case follows as nicely. The cycle index in this case is slightly more

complicated.

Theorem 1.2, implies that we can enurnerate a larger collection of circulants using

Polya's theorem and Turner's clever method. Thus, we can enumerate the circulant of

order 8,9, 18 and 2'm where e E {O, 1: 2) and m is odd and square-free. Here we consider

only the case where rn is odd and square-free. The forrnulae for the remaining values will follow fiom the results fiom this and the next chapter.

Working with the group of units of Zn when n is not prime is not as straightforward

as the prime case. However, we c m show that a group of units is as straightforward as

working with product of cyclic groups.

3.1 LEMMA. Let U ( R ) denote the uni& ofa ring R I /S = RI CI? R2 63.. . $ &,forrings

R;, 1 5 i 5 t , with multiplicative identities 1 R,, then

U(S) = U(R1) x U(R2) x . - . x U ( R t ) .

PROOF: [14] Addition and multiplication are defined in the naturai way, that is, component-

wise. We have that

CHAPTER 3. CIRCOLANTS AND CIRCULANT DIGRAPHS 19

Thus, (q , 7-2, . . . , r t ) has an inverse if and only if ri has an inverse for each i fiom 1 to t.

Thus the group of uni& of RI $ RÎ $ . . .6 Rt is precisely U ( R l ) x LJ(R2) x . . . x U ( R t ) .

3.2 THEOREM (Chinese Remainder Theorem). Suppose rliot m = ml m, . . . mt and fhnl

gcd(m;,rn,) = 1 for i # j. Let b1,b2<.. . <bt be integers and conrider the systern of

congnrences:

z = b i ( m o d n i ) , f o r i = l ... t. .

There exisîs a unique solution to thispmblem. modulo m.

The Chinese Remainder Theorem allows us to prove a well known result.

3 3 THEOREM. Let rn = mlnz ... ml, where the m, arepairwise CO-prime. Then

PROOF:[14] Let 9; denote the natural homomorphism x ct x (mod mi) fiom Z to Z,,

for i = 1, . . . , t . These form a natural map ~ from Z, to Z,, @ Z,, $ . . . @ Z,, defined by : $(n) = (QI (n), &(n), . . . , $t (n) ) for al1 n E Z,. This is a ring homomorphism

as each +i is itself a homomorphism. Further, it is an isomorphism because the Chinese Remainder Theorem guarantees a well defined inverse fûnction. The inverse of 11 takes

(x 1. xz , . . . , z,) to the unique solution modulo rn of z z; (mod mi), i = 1 . . . t . This

gives us an isomorphism between Z, and Z,, @ Z,, $ . . . @ Z,, and nom Lemma 3.1, z; s z;, x z;, x . . . x zm,.

3.1 Circulants

We first consider circulants that have an order that is a product of two distinct primes. It

will become clear how to generalize this to the larger case. Circulants of the fom X(W; S)

CHAPTER 3. CIRCUZANTS AND CIRCULANT DIGRAPHS 20

with p, q prime cm be enumerated in a manner similar to the circuiaats of prime order. Each circulant will be associated with a function fiom

1 {i,-i} E S

O otherwise.

Theorem 1.2 tells us, that as in the case of prime order circulants, the group action to

consider is multiplication by an element a in Z,. Functions, and consequently connection sets, which can be rnapped to each other under this action yield isomorphic circuiants.

As before a E Z, acts on D by mapping {i, -i) to {ai, -ai). Thus a and -a per- form the same action. Hence, the group of permutations we consider is Z;,/{l, -1}. By

Theorern 3.3 this group is isomorphic to (a; x Z;)/{(l, l), (- 1, - 1)) which we denote by

The cycle index is calculated by examining the action of the permutation group on the domain.

3.4 THEOREM. The cycle index of P,, acting on D is

a a is odd

a /2 otherwise

and

lcm(dl , d 2 ) / 2 dl, d2 60th ewn and divisible by the same power of 2 L(d1, d2) =

lcm(di , d2) otherwise.

CHAPTER 3. C I R C U ' S AND CIRCULANT DIGRAPHS 21

PROOF: Let us denote the element {m. - m ) of D by m , with O < rn 5 7212. An element of

Pp, is of the form {(a y b) , ( -a , - 6 ) ) . We cm represent it uniquely by g(,,b), with O 5 a < p

and O 5 b 5 q / 2 . We have that g[.,b) acts on rn in the following way: g(a ,b l (m) = z where

x r am ( m o d p ) , x G h (mod q ) . The Chinese Remainder Theorern guarantees a unique

value for x. Notice that g c , b ) ( m ) = z where z = a2m (mod p ) , x G b2m (rnod q ) and

in general, gf, ,b)(rn) is the solution to z G a'm ( m o d p) , x = bim (mod q ) . Also notice

that if m is a multiple of p this system reduces to z = O (mod p ) , z = b'm (mod q ) .

Thus multiples of p are mapped to other multiples of p. In fact we can partition D into

three parts, multiples of p, multiples of q and those coprime to pq. Since a multiple of p is

necessarily coprime to q, there are O ( q ) / 2 members in the first part, iP(p)/2 members of

the second part, and @ ( p q ) /2 members of the third part.

To determine the cycle index we must find the cycle structure of each permutation.

Consider m E D. We have already determined that the permutations permute m within the part to which it belongs. Hence, lets consider the action of a general permutation on rn and

consider the three parts separately. Let ka = ka and kb = ord,(b).

First, let m be a multiple of p. Let g(o,b) be a general action fkom P,,. The length

of the cycle in which rn is contained under the action g(.,b) is the l e s t k for which either 9fa ,b , (m) = m or gfa ,b)(m) = -m. niat is, the least k for which bk = 1 (mod q ) or

bk I - 1 ( m o d q ) respectively. The second case occurs exactly when ka is men, and in this

case k is k a / 2 . In the first case k = ka. There are O ( q ) / 2 = (q - 1) /2 multiples of p in

D, and each is contained in a cycle of length k . This will have the e f f d of contributing an 4-1

x k to the cyclic monomial of g(.,b). The w e for multiples of q is identical. There are cycles of length k , where k is kb if kb is even and k is kb otherwise.

If rn is coprime to both p and q, then we mut contend with the double equivalence.

The element m is contained in a cycle of length k , where k is the smallest integer such

that gtaVb, (m) = rn or gf, , , ) (m) = -m. The first case occurs when a' i 1 (mod p), and

bk = 1 (mod q ) . The smallest k for which this can happen is k = 1cm(ka, kb ).

The second case happens only when a' = - 1 (mod p) , and bk = - 1 ( m o d q ) . From

discussion on multiples of p that would irnply that this case occurs when kb and ka are both

C W T E R 3. CIRCULANTS AND CIRCULANT DIGRAPHS 22

even, and hence k = lcm(k,, kb) /2 . However, if Icm(ka k )

kh 1 lcm(k., kb) /2 then b- = 1 (rnod q ) . Hence, this case occurs exactly when kb and ka are both even, and neither kb llcm(ka, kb ) /2 nor ka llcm(k,, 4 ) / 2 . The latter two occur when one of kb and ka contains a power of two higher than the other. In the Iatter two cases, k = lcm(ka, kb ).

So there are @ ( p q ) / 2 = ( p - 1 ) (q - 1 ) /2 menibers of this part, each contained in a

cycle of iength k , thus ~Ilderg(.,~) there are ( p - l ) ( q - 1 ) / ( 2 k ) cycles of kngth k.

Let ka = ord,(a) , and kb = mdp ( 6 ) . Putting al1 three cases together,

obq) p-le Icm(ka.kb) 2ka 2kb

x l - ( k a . k b ) / 2 x k a / 2 x k b / 2 ka even, ka even and divisible by the same powers of 2

Now that the possible cycle actions are surnmarized, it remains to determine the number of permutations of each cycle type. These can be counted by enumerating through the

possible orders of elements in Z, and Z,.

To begin, consider the following observation.

3.5 LEMMA. The multiplicative order of -a in Z, p prime. dependr on the midue of

ordp(a) rnodulo 4. Ifordp(a) is congruent to O rnodulo fou,: then ord,(-a) = ordp(a). If

ordp(a) is odd. then ordp(-a) = 2ord,(a). OtherwiSe, ord,(-a) = ordp(a) /2 .

PROOF: Notice that if ordp(a) = k ( a ) , then (-a)2k(") (-1)2k(")a2k(a) G 1 (mod p).

Hence, ordp(-a) 12k(a). By symmetry, k ( a ) 120rdp(-a). Thus, ordp(-a) E { k ( a ) , y, 2k(o)) .

C W T . 3. CRCULANTS AND CIRCULANT DIGRAPHS 23

Let k ( a ) be divisible by four. Then, ( -a )k (a ) /2 = ( -1)k(a) /2ak(a) /2 r - 1 (mod p).

Hence, ord,(-a) = k ( a ) . Nexf if k ( a ) is congruent to 2 modulo 4, then ( - u ) ~ ( " ) / ~ =

( - ~ ) ~ ( " ) / ~ a ~ ( ~ ) / ~ E 1 (mod p ) , and the order of -a is k ( a ) / 2 . Lastly, if k ( a ) is odd, then ( - a ) k ( a ) = ( - l )k (o )ak(o) -1 (mod p), hence the order of -a is 2k(a) .

Now, the number of elements of order k modulo p is @(k). Lemma 3.5 gives us a way to

count the number of elements a of order k modulo q where O 5 a < q/2. Now, recall that

if x E Z,, is in a cycle of length 2k under multiplication by a , the corresponding element

of D, { x , - x ) , w i U be in a cycle of length k under the permutation associated with a in P. I f x is a multiple of q, this cycle length will be ka = ord,(a) . Now, if ka is divisible by 4,

-a will result in a cycle of the same length. Hence, there are @ ( k a ) / 2 = @ ( k a / 2 ) elements

between O and q/'Z which will put an element in a cycle of order ka. If ka is even, but not

divisible by 4, then it will still put {x, -x) in a cycle of length k a / 2 . in this case the order

of -a modulo p is kJ2, so will also put the element of D in a cycle of length kJ2 . Thus, the number of elements between O and q/2 with order ka is @ ( k a / 2 ) .

Just by considering only even orders, we can account for al1 of the orders of elements

between O and q / 2 .

We have that

CHAPTER 3. ClRCULANTS AND CIRCULANT DIGRAPHS 24

We can split the final surn up into different cases based on the parity of the fhst divisor.

3.6 COROLLARY. Let p, q beprime. The number of cimlant graphs of order n = pq up to

PROOF: This value is achieved by substituthg 1 RI = 2 into every value of x.

Consider now some examples for some small primes.

EXAMPLE. We can calculate the number of circulants of order 6. According to Theo-

rem 3.6, this number is 23 = 8

since the cycle index is x t , which is evident fiom Figure 3.1.

EXAMPLE. We c m calculate the number of circulants any large suitable orda. The number

of graphs of order 35 is 11 144. W e can also detemiine the number of each valency. R d 1 that to determine the generating nuiction where the weight is valency we simply substitute 1 + u2' for xi in the cycle index. In this case we get,

CHAPTER 3. ClRCULANTS AND ClRCULANT DlGRAPHS

Figure 3.1: THE EIGHT ISOMORPHISM CLASSES OF CIRCULANTS OF ORDER 6

We can generalize the enurneration formula with some help nom notation. Let 1cm({ai))

denote the least cornmon multiple of a set. That is, for each a; E {a), a;llcm({a)), and

it is the smallest integer with this property. I f the set consists of a single element a, then lcm(a) = a.

3.7 THEOREM. Let pl, p2, . . . , p, be a collection of distinct, oddprimes. The cycle index of

the group action of Z; /{l , - 1) acting on Z ,,,--,, /{1, - 1) \ O by nghr multiplication

is

where H and L am as in Theorem 3.4 und P ( I ) = n i E r ( p i - 1).

PROOF: The proof is very similar to Theorern 3.4. Theorern 1.2 holds in this case and we use the same isomorphisrn and action fiom the group to the set. An element of

Z,,, / (1, - 1) \ O can be represented by m with O < m < pl pz . . . pt 12. An element of the SouP a E ZPl,.-,g is represented by g(,,,a,I...I.,) = g~ with ai a (mod pi) and O 5 a, 5 p, /2. Each go acts on rn as g, (m) = x where z is the unique solution modulo

CHAPTER 3. CIRCULANTS AND CIRCULANT DIGRAPHS 26

plp2 . . . pt to the system of congruences z = a i m (mod pi) . nius, g:(m) is the solution to

z = a f rn (mod p i ) and in general g,"(rn) is the solution x to x = afin (mod pi).

In the case of t = 2 three cases were aident. The possibilities were either that m

is coprime to both pl and p2 or to exactly one of pl or f i . In the general case, m may

be coprime with some subset of the pi. The length of the cycle in which rn is containeci

will depend on this subset. Say without l o s of generality îhat m is coprime with exady

pi. p2 ' . . . ,p .> for 1 5 s 5 t. Consider the action under 9.. For i > s, a ; m = O (mod pi) .

This means if k = Icrn({ordp , (a) ) ) , i = 1 . . . s, we have gt(rn) = m. Thus, the cycle

length of g, divides k. This will be the order of the cycle unless there exists a k' for which

&(m) = -m. Such a k' < k exists, as before, if and only if, each ordPi (a) is even and not

al1 are divisible by 4. If such a k' exists it is kt = lcm({ordp, ( a ) ) ) / 2 .

It is not difficult detennine the number of rn coprime with exactiy the set { p l , n, . . . , p,),

or indeed any subset of the p; in general. There are @ ( p i ) elements of Z,,,..,c coprime

with pi, hence there are n:==, @ ( p i ) /2 possible values for m, each coprime with exactly

PlP2 - - - p , -

As before, we sum over the possible orders to achieve the result.

3.8 COROLLARY. The number of cimlants of order pl pz . . . pt with each pi a distinct odd prime is

3.2 Circulant Digraphs

The digraph case is simpler. As we are not looking at the quotient group, by merely de- termining the cycle index of Z:, acîing on Zn, we cm count the classes. This rernoves the

awkward terms that result fiom daermining the parity of orders. Though, in r d i t y in the

circulant case we have hidden this awkwardness in notation.

CHAPTER 3. CRCULANTS AND CIRCULANT DIGRAPHS 27

3.9 T H EO REM. Let p l , pl, . . . , p, be a collection of distinct, odd primes. The cycle index

of the pvup action of Z,l,-.,c acting on Z,,,..,, \ O by lefi multiplication is

where L is as in Theoriem 3.7.

3.10 COROLLARY. The number of digraphs of order p1p2 . . . pt , . with each pi prime, i =

EXAMPLE. The number of non-isomorphic circulant digraphs of order 15 is

Chapter 4

Unit Circulants

The question remains, what of the circulants of other orders? If we are relying on this tech-

nique, we can Say nothing of the orders which fall outside of those that satisfy Muychuk's

Theorem. However, the recent proof [8] of the following conjecture of Toida [1 SI allows us

to continue in the same fashion to count a family of circulants with no restriction on order.

4.1 THEOREM (Dobson, Moms). For any S C Z,. whenever X ( n ; S') LF isomorphic to

X ( n ; S ) . the= exists an a E Z:, satisfying S' = as.

This result allows us to enurnerate the circulants X ( n ; S) of order n whose Cayley

subset is a subset of Z;. We shall cd1 such a circulant a unit circuhnt. Notice that the

circulants of prime order are al1 unit circulants. We cm also defhe the class of unit cimulant

digraphs. These are the digraphs with the analogous property, that is, circulant digraphs of

order n with the connection set a subset of Z:.

EXAMPLE. If n = 6, then the Cayley subsets which yield unit circulants are the empty set

and {1,5) = Zé, as depicted in Figure 4.

In the mode1 we have used so far we use the domain of Zn in the case of digraphs

and Z;/{l, -1) in the case of graphs. 'Ibeorem 4.1 implies that the permutation group

to use in Polya's Theorem is Zn. Hence we are essentially looLing at Z i acting on itseif.

To use Polya's enurneration result we need to determine the cycle index of this adion.

CHAPTER 4. UNIT CIRCUZANTS

Figure 4.1 : THE COMPLETE FAMILY OF UNIT CIRCULANTS ON 6 VERTICES

The difficulty lies in detennining the order of an arbitrary elexnent in Z;. Forhmaîely, the

structure of Z:, is well studied and we can make use of an isomorphisrn of Zn to a product

of cyclic groups to determine the cycle index.

4.1 Odd Prime Powers

We know from Theorem 2.8 that Z; is isomorphic to the cyclic group of order p - 1 when p

is prime. To determine the the group of units for other orders, we will require some number theoretic tools.

DEFMITION. An integer a is cailed a primitive mot rnod n, if a generates the group Zn.

Equivalently, a is a primitive root rnod n if a is of order O ( n ) modulo n. If there exists

some primitive root modulo a given n, then we have that Z i is cyclic.

EXAMPLE. TO see that 2 is a primitive root rnod 5 notice 22 e 4 (mod 5 ) , 23 = 3 (mod 5 ) ,

and z4 = 1. On the other hand, 8 has no primitive root as 32 = 52 E T2 n i (mod 8).

4.2 LEMMA. ( f t > 1 a n d a i b (mod then aP = 6P ( r n o d p t f l ) .

4 3 COROLLARY. I f t 2 2 and p # 2. then ( 1 + ap)pC-' E 1 + apt-l (rnod p t ) for al1

a E Z. Furthe>: i f p does not divide a, then pt-' is the order of 1 + a p modulo pt.

PROOF: We can prove this result by induction on t . The result is trivially true when t = 2.

Next, Say it is true for 2 5 t = n. Then we have

( 1 + a p ) ~ n - 2 1 + apn-' (mod pn)

( 1 + a p ) ~ " - ' (1 + apn-' )P (mod pn+l ) by Lemma 4.2

1 + (7 ) apn-' + B (mod prit')

where each term in B contains P"+' as a factor since it contains'a power of pn-L greater than 1. Hence, the result holds for t = n + 1 and the result follows by induction. Now, this

implies that ( 1 + ap)pt-' 1 (mod p t ) , since the order of 1 + ap divides pt-'. However,

(1 + ap)pC-' 1 + apt-' (rnod pt), and hence as p does not divide a , this is not 1. Thus the order of 1 + ap is greater than pt-*, and hence is pt-' .

4.4 THEOREM. I f p Lr an oddpnme and t E Z +, then Z,. is cyciic.

PROOF:[14] To prove the result, it is sufficient to establish the existence of an element

of Z,, with order 8 ( p t ) = (p - l ) p t - l . The case t = 1 was proven in Theorem 2.8,

hence we may choose an element x E Z p with order p - 1. If xp-' f 1 (mod p2), then we can, and momentarily shall, illustrate that x is the desired element of order O ( p t ). If

xP-' = 1 (mod then x + p is also an element of order p - 1 rnod p and

The expression (p - l ) x ~ - ~ is clearly not divisible by p, thus (x + p ) ~ - ' f 1 (mod

We can assume without loss of generality that xp-l f 1 (mod P~). We can write xp-'

as 1 + ap, where p does not divide a. The multiplicative order of 1 + ap modulo pt is pt-l

b y Corollary 4.3.

Now, consider any whole number n such that xn = 1 (mod P'). Thus, (xn)p- l

( 1 + ap)" = 1 (mod pt ), and must divide n. Further, xn = 1 + bpt for some intega

CHAPTER 4. UNIT CIRCULANTS 31

6, so xn = 1 (mod p). If we write n = ~ ' - l r n , then since XP = x (mod p), 1 E 2" E

x* (mod p ) . Since the order of x modulo p is p - 1, p - 1 divides m. Thus, for any n

such that X* G 1 (mod pt ), ) divides n. We have illustrated a primitive root modulo

pt , implying that Z,t is cyclic for prime p.

It remains to determine the cycle index of Z;, on itself. Once we have established this result, the enurneration formula will fa11 from it. From this point on assume that D and R

are as they have been to this point.

4.5 THEOREM. gp be is an odd prime, then

PROOF: The goup action of a E Z;, acting on itself is a, (x) = ax (mod pt ). Hence, an

arbitrary element x E ZPt is contained in the cycle

where k is the multiplicative order of a in Z,t. The size of Z,, is hence each a 9g

contributes a terrn of x, .

The fact that the group is cyclic irnplies that there are O ( k ) elements of orda k in the

group.

4.2 Products of Odd Prime Powers

Next we consider the product of odd prime powers.

CHAPTER 4. UNIT CIRCULANTS 32

4.6 THEOREM. Let p and q be distinct odd primes and let n = pr q'. The cycle inder of Zn

acting on ifselfis

PROOF: We have so far that the automorphism group is isomorphic to Z,, x Z;.. Given

(a, b ) E Z;r x Z ;,, the corresponding action g(=,b) on Zn maps m E Z; to the unique solution x modulo n of

x r a m ( m o d p r )

x = h (mod qs) .

It fotlows that gfo,b)(m) is the unique solution x modulo n of

x aim (mod pr)

x bim (mod q').

Hence, the general element m E Zir,. is contained in the cycle

where k is the smallest integer such that m = mak (mod pr ) and m dk (mod qs ) .

It must be that k = Icm(k(a) , k(6) ) . We have already illustrated that there are O ( k ( a ) )

elements of order k ( a ) in ZPr, thus the number of pairs (a, 6 ) with a, b of ordas k ( a ) , k(b)

respectively is O(k(a ) )O(k (b ) ) .

4.7 COROLLARY. Ifn = pllp;l . . . pir whem each pi is a distinct oddprime, then

4.3 Unit Circulants of AU Orders

Powers of two are only slightly more complicated to incorporate.

4.8 THEOREM. Primitive mots erî« modulo 2'for t = 1 and 2. hence Z2 and Z4 are cyclic.

if t > 2. then Z;, S Z2 x Z2t-2.

PROOF: 1141 The fmt statement follows fiom the observation that 1 is a primitive mot

modulo 2 and 3 is a primitive root modulo 4. Henceforth let t > 3; We show the equivalent

statement that A = {(- l)"jb la = 0 , l and O 5 b c 2t -2} is a reduced residue system

modulo 2'. That is, every element in Z> is equivalent to an element in A modulo 2'. We

prove by induction that

52t-3 = 1 + 2'-L (rnod 2'). (4- 1)

This is clearly true for t = 3. Now assume it is true for t = n. We have,

52n-3 G 1 + Y-' (mod 2") - 2-2 * a = - (1 + (mod 2"+') by Lemma 4.2

= 1 + 2" + 22n-2 (mod 2"+')

1 + 2" rnod 2"+l

since 272 - 2 2 n + 1 when n 2 3. We have established (4.1) by induction. In proving the

daim we have established the multiplicative order of 5 in Z2t to be 2'-2.

We next show that the members of A are distinct in in Zzt. If they are distinct, A will

cover al1 of Z,t since we have already discovered an injective relationship.

Say that -sb e 5" (mod 2'). That would imply that

gb-b' i -1 (mod 2')

2(b - b') = O (mod 2'-*)

(b - 5') = 2t-3 (mod 2'-*) 52t-3 n -1 (m0d 2')

contradicting (4.1). I f sb G sb' (mod 2' ), then

5 6 - b f E 1 (mod 2')

( b - b') = O (mod

* b = b'.

Since ((-1)a5b)2t-2 = 1 (rnod 2'), no elwent in Z;, has order higher than 2t-2, and thus there could be no primitive mots of 2', for t 2 3.

We can now close the story on the unit circulants and unit cirdant digraphs. Theo- rem 4.1 establishes the comection between the permutation group of the comection sets

and isomorphism classes as was the case for circulants and circulant digraphs of prime

order.

4.9 THEOREM. Let n = 2'1 p;' . . . pit whem each pi is a distinct O& prime. The number of unit cimlants of order n is

when rl 3 3,

when O < rl < 3, and

PROOF: We proceed in a rnanner identical to the enurneration of circdants of prime order.

n i i s is justified by Tnieorem 4.1. A substitution of 2 into each xi of the cycle index gives the result.

4.10 THEOREM. If n = 3'1 p? pjJ . . . PL', where each pi is a distinct odd prime, then the

number of mit cimlant digraphs of O& n is

when r 2 2, and

To illustrate these, it is best we end this section with a couple of examples.

EXAMPLE. Consider the unit circulants of order 1 6. The formula yields

as one can see fiom Figure 4.3.

EXAMPLE. We can use the formula fiom Theorem 4.9 to count the number of unit cucu-

lants of order 35 to compare with the number of circulants. The number of unit circulants

This is a srna11 portion of the 1 1 144 total number of cirdants of order 35.

Figure 4.2: NON-ISOMORPHIC UNIT CIRCULANTS OF ORDER 16

By way of contrast we can also calculate the number of unit circulant digraphs of order

35. The nurnber of non-isomorphic circulant digraphs of order 35 is

Chapter 5

Cayley Graphs over Z p x Z p with p

prime.

The strategy of appealing to Polya's Theorern to count isomorphism c1as.w can be used to

count the Cayley graphs on Zp x Z,, since this is another family of CI-groups. However, the automorphism group of Z, x Z, is the general linear group and hence our methods to

deterrnîne the cycle index are quite different than for detemiinhg the cycle index of a cyclic

F'uP - We consider Z, x Zp as an additive group, and co~l~equently think of Zp x Z, as a two

dimensionai vector space over Z;. In generai we consider of ZF to be an n - dimensimal

vector space over Z,. With this view in mind, the group of automorphisms is clear.

5.1 THEOREM. The automorphism p u p of Z," is isumotphic to GL(n, p) . the gmup of

invertible n x n matrices over 2,.

PROOF: Define e; = (0,. . . ,O , 1,0,. . . , O), with the 1 in the i th position. Since Z," is generated by the set of e;, l 5 i 5 n, the action of any automorphism is determinecl by its action on {e i , i = 1 . . . n).

We can cunstnrct a general automorphism a, and in the process count the number of

possible automorphimis. We have pn - 1 non-zero elements to which we can map el.

CHAPTER 5. CAYLEY GRAPHS OVER Zp x Zp WTH P PRIME. 38

Our only constraint in detennining the value of a ( e z ) is linearly independence with el, and

hence none of the p scalar multiples of a(el) leavhg pn - p possibilities. Likewise, the

choice for a(e3) cannot be in the span of &(el) and a ( e z ) , thus pn - pz possibilities. in general, there are pn - choices to which one may assign a(ei). Thus the total number

of possible automorphimis a of the group Z," is

which we shall denote [ p 1,. Now, clearly the action of any A E GL(n , p) is an automorphism of Z,̂ and hence

GL(n, p ) C Aut(Z,"). As the sizes of the two sets are equai, the two sets are equal.

The problem of deteminhg the cycle structure of linear transformations over a finite

field was fvst tackled by Kung in 161. His aim was to determine characteristics of random

matrices. His main tool was a vector space analog of the Pdya cycle index, hence we will

require effort beyond his work. Recently, Fripertinger [3] calculated the Pdya cycle index

of the general linear group (as well as afke and projective groups) and used the cycle index

to enmerate isometry classes of linear codes.

The cycle index as Fripertinger calculated it is suitable to enurnerate digraphs, but re-

quires modification to be usehi for the undirected graph case. Furihermore, since we are first interested in specifyuig n = 2, we can &te the expression in a far more explicit,

though less compact, form.

5.1 Rational Normal Form

In this context we are regarding the matrices as permutations. Recall that permutations which are conjugate have the same cycle structure. This is the essence of our strategy. As

each matrix is in a single cmnjugacy class, we can detennine the cycle index by detemiining

CNAPTER 5. CAYLEY G R A P H S OVER Z p x Zp WITH P PRIME. 39

the size of each conjugacy class and the cycle structure of a representative. This section

defines the rational normal form and illustrates its suitability as a representative.

Let V be a vector space of finite dimension n over field F . An automorphimi is a

bijective linear transformation of V to itself. Let A be such a linear transformation of V

over F throughout. The notation iFp indicates the finite field of p elements. The vector

space we will consider is Z p, hence, we have -4 E GL(n, p) . However, these results hold

over any field and hence we shall present them in full generality when this is reasonable.

DEFINITION. The space V is cyclic with respect to A if there is some v E V such that

{ u , .4(v), A2(v), ..., .4n-1(v)) forms a basis for V.

DEFINITION. A polynomial #(x) E F [ x ] is an annihilatingpofynornial in V of A if

and only if d (A)v = O for every u E V. W e cail 4(t) the minimal polynornial of A if

d(x ) is the monic annihilating polynomial of minimum degree. A is guaranteed to possess

an annihilating pol ynomial, and consequentl y a minimal pol ynomial, since the Cayley-

Hamilton Theorem states that the characteristic polynomial det(A - Ix), of a matrix A is

annihi lating .

DEFMITION. Let Q ( x ) = a0 + alz + . . . + a,xr E IF[x]. The companion rnahir of d(x)

is the r x r matrix

C(d) =

Notice that the minimal polynomial of C(6) is 4 ( x ) .

If V is an n-dimensional vector space cyclic with respect to linear transformation A and

the minimal polynomial of A is +(x) = a0 + alz + . . . + a,-lzn-l + zn, then there is a

ba i s of V over F such that in this basis the matrix of A is C(4). The rational normal fonn

is a generalization of a companion matrix.

A vector space V can always be broken down Uito cyclic subspaces. R d 1 [5] that

given a linear transformation A of V over F, with minimal polynornial +(z) = ni + i ( ~ ) ~ ' ,

CHAPTER S. CAYLEY GRAPHS OVER Zp x Z p WlTH P PRIME. 40

with each & ( x ) monic, irreducible, and unique, then A decomposes V into a direct sum of ni-dimensional cyclic subspaces, each invariant under di (2)'s for some i. Notice that

Ci n;f; = n.

DEFINITION. Theprimary decomposition of A is a unique representation of V as a direct

surn of A-invariant subspaces Lii such that O-* is the kernel of 4; (A) '* .

Each A-invariant subspace Lii can be fkher decomposed into a direct sum of subspaces

Ci,,j S U C ~ that -4 restncted to U i j is cyclic. Each C i i j is the k d of @i(A)J for j 5 ci.

DEFINITION. A partiiion of n is an unordered set of integers which sum to n. We can

summarize a partition with a partition vector A = (A1, A2, X3, . . . ), a sequence of non-

negative integers with finitely many non-zero tenns such that n = A . ( 1,2 ,3: . . .) = 1 Al + 2X2 + 3A3 + . . . . Henceforth partitions will refer to the vectors and [A I will denote the size, which is n.

DEFRIITION. Given the r x r cornpanion matrix C(4) we can define the associated k r x k r

matrix hown as the hyperrompanion ma& &(") by

where Eir = (eij);<,,jSr with eij = 1, if (i, j ) = (1, r ) and O othenuise, and 0, is the r x r

O-matrix. Notice that d(') = C(4) and that the minimal polynomial of 4(k) is d(z)("*

DEFMITION. Given a monic, irreducible polynomial 4 ( x ) of degree d , and a partition A

we define the d 1 X 1 x d X 1 maîrix D ( 4 , A) as the diagonal block matrix

5.2 THEOREM. Suppose A is a Iinear transformation of an n-dimemional rpace wàth rnin-

imal polynomial $(z) = < P ( x ) ~ , with $(z) monic, i duc ib l e and of deme d. Them erists

a parîition X with 1 X Id = n such that A is zhilar to D(4 , A) .

CHAPTER 5. CAYLEY GRAPHS OVER Zp x Zp W7H P PRIME. 41

The partition cornes from the way A decornposes V into cyclic subspaces. That is, X i is

the number of cyclic subspaces of V of dimension i.

5 3 COROLLARY. Let A be a linear ~nsfonnafion in the n-dimensional vector space V

over F with minimal polynomial #(x) = ni=, + i ( ~ ) c s with each 4; unique. monic and

irreducible and di the degree of 4,. Them eristx a sequence of parîiti0n.s (A( '1, . . . ) with Ci 1 A( '1 1 di = n, and an ordered basîr of V such that A dative to thut busis LF

DEFMITION. The matrix A of A as described in the above corollary is the mtional n o m l

fonn of A. Each matrix is similar to a matrix in rational normal form unique up to the ordeïing of the blocks.

To detennine the rational normal form for a matrix A, first factor its xninimal polyno-

mial +(x) into d(x) = ni o i ( ~ ) c s where each 4; is M irreducible minimal polynomial. Determine the primary decomposition of V into Ul 8 @ - @ Ut where Ui is the kernel of @:(A). For each i determine the number A" of subspaces of Cli with dimension qnj

which are cyclic with respect to the restriction of A on Ui. This will correspond to the num-

ber of spaces which are contained in the kernel of QJ (A) but not the kernel of $n~"l-L(A). The sum of al1 dimensions of d l cyclic subspaces must total n, the dimension of V, that is,

Ci Ix(') Id; = n.

EXAMPLE. TO illustrate this process consider the following A E GL(3 ,5 ) . Let A = 1 1 1 1 2 3 ) . The minimum polynomial of A is (x - 2)2(2 - 4). The space spanned by (A - 213)*

( 2 3 0 1 2 0

is equd to the space spanned by ( 8 8 g ) . and hence the kemel is two dimensional. The

kemel of A - 2 I3 contains only the zero vector since it is of full rank. Hence, the kemel of

( A - 2 13)* is cyclic. Thus, A(') = (0 , l ) . On the other hana the kernel of (A - 4 1 3 ) is one dimensional and so Ac2) = (1). This gives a rational normal form of

CHAPTER 5. CAYLEY GRAPHS OVER Z p x Zp WTTH P PRIME. 42

EXAMPLE. To construct au n x n matrix in rational normal form one needs only a col-

lection of irreducible polynomials +1, &, . . . ,4< of degrees d l , d2, . . . , d, each l a s than n,

respectively, and a set of t partitions A(') such that Ci IA(')ld, = n. When n = 2 it is not

a sîrain to imagine al1 of the possibilities. Let V = Z, x Z,. We will wnsider the various

definitions and consequences in this context.

The irreducible polynomials of degree at most two are of the form x2 + az + b and z - a,

witha,b E Z,.

If a matrix has an irreducible, degree two, minimal polynomial #(z) = x2 + ax + b,

then clearly al1 of V is annihilated by 4(A). Thus, the partition X(l) = 1, since there is one

cyclic subspace of dimension 2 1. The rational normal fom is

If the minimal polynomial factors as dl (x)& (x) = (x - a) (x - b), then the primary decomposition will split V into (II eu2 where CIi = kernel( A-a 12) and LJ2 = kernel (A-

b l z ) . These are both of dimension at least one and since the dimension of V is 2 there are

both of dimension exactly 1, and hence cyclic. Thus, A(') = (1) and A(') = (1). The

rational normal fonn is

Next consider x - a. There are two possible partitions of 2, (2) and (0,l). The first

implies that there are two cyclic subspaces of dimension 1 hence A - alz annihilates ail of

V and x - a is the minimum polynomial. This gives rise to a rational normal form of

In the other case, there is one cyclic wbspace with respect to the matrix, of dimension

2. Thus, (x - a)2 is the minimum polynomial and the rational normal fonn is

CHAPTER 5. CAYLEY GRAPHS OVER Zp x Z p WTH P PRIME.

5.2 The Size of a Conjugacy Class

Kung [6] detennined the sïze of a conjugacy class in GL(n, p). Let d(x) E F, [z] be a monic, irredufible polynomiai of degree d, and let A = (A, , X2, . . . ) be a partition.

5.4 THEOREM ( [6]) . The size of the centralirer of D(q5, A ) in GL(lAld,p) is

A key thing to notice here is that the number depends on d and the partition, not the poly- nomial itsel f.

5.5 COROLLARY. The number of maîrices A f GL(n, p) wirh rational nonnal fonn

PROOF: Every ma& is conjugate to a unique ma& in rational normal fom. As conju-

gacy defines an equivalence relation, to count the number of matrices with a given rational

normal fonn we can determine the size of a conjugacy class. The cardinality of a conjugacy

class in a group G containing a fixed element g is the order of G divided by the order of

the centralizer of g in G. It is thus sufficient to show that the order of the centraliza of

A, a rnatrix in rational normal form, is n:=, b(A('), d, p). It is a block matrix hence the centralizer is the direct product of the centraiizers of each block. The order of a block is

given by Theorem 5.4.

CHAPTER 5. CAYLEY GRAPHS OVER Zp x Zp UflTH P PRIME. 44

EX AMPLE. Again we examine the n = 2 case to chri@.

Consider the numba of matrices with rational normal form D [ x - a, (2)]. These are

the scaiar multiples of the identity. We would expect exactly one each of these since these

matrices comrnute with any other and are the only matrices with produce an effect of scalar

multiplication. We calculate p I = 2 , and b ( ( 2 ) , 1, p) = (p2 - p ) ( p 2 - p ) . This is exactly

[ p l2 hence the number of matrices in a conjugacy class with D[z - a, ( 2 ) ] is exactly one.

Similarly we can calculate the vdues for the other classes and tbey work out as follows.

Minimal pol ynornial Rational normal form Number in a class

Table 5.1 : Sizes of Conjugacy Classes in GL(2, p)

5.3 The Cycle Index of G L(2, p )

The cycle index of G L ( 2 , p) acting on Z, x Z, \ (0, O) is nearly at hand. The previous

examples have illustrated the complete set of conjugacy classes and theu sues. To finish

we require the cycle structure of each class acting as a permutation.

Observe that in general, the action of an n x n diagonal block matrix on Z," can be

decomposed into a product of actions of the blocks on vectors of the appropnate length. Now, Poiya observed [I l ] that given groups G and H acting on sets X and Y, respectively,

the cycle index of the induced action of the product G x H on X x Y, (g, h) ( z , y) =

(91, h y), can be expressed

2 ( G x H ) = Z ( G ) Z ( H ) .

A rnatrix A in rational normal f o m is a block diagonal matrix. Each block is a hy-

CHAPTER 5. CAYLEY GRAPHS OVER Z p x Zp WITH P PRIME. 45

percompanion matrix of a monic, irreducible polynomial4i of degree di over Z,. We can decompose this into an action of the direct product of hypercompanion matrices,

acting on

Hence the problem of detemiining the cycle index of any matrix reduces to determining the

cycle structure of hypercompanion matrices of monic irreducible polynomials.

To determine the cycle structure of just such a ma&, consider the co~ec t ion between

the hypercompanion maîrix and its polynornial.

DEFINITION. Given A f G L(n , p), we define the order of A to be the least integer k > O

such that = In, the n x n identity matrix.

DEFMITION. Let 4 E F, [XI be a polynomial of degree m such that +(O) # O. The order

of 4, denoted ord(4) or ord(d(x)), is the least integer k > O such that $(x) divides xk - 1.

It cm be shown that sorne k 5 pm - 1 exiStS. If 4(0) = O, we cm define the order of q5 by

writing 4(x) = xn $(x) such that n E Pi, $(O) # 0, and define ord(4) = ord($).

5.6 THEOREM ([7]). Let 4 E F, [z] be an irreducible polynomial over Fp of degree n with

Q ( 0 ) # O. Then ord(q5) Lr equal to the onier of any mot of 4 in the multiplicative gmup

5.7 COROLLARY ([7]). Let E F, [XI be an inducible polynomial over Fp of degree m

with +(O) # O. Then ord(q5) divides pm - 1.

The following provides a very usefbl summary regarding the number of irreducible poly-

nornials of a given degree.

CHAPTER 5. CAYLEY GRAPHS OVER Zp x Zp WTH P PRIME. 46

5.8 THEOREM ([7]). The number of monic imducible pofynomials in Fp [x] of degree m

and order k is equal to <P(k)/rn if k 2 2 and m is the multiplicative order of p modulo k.

It is 2 ifm = k = 1 und O in all other cases.

If m = 2, then for al1 positive k such that k [ ( p 2 - 1) and p $ 1 modulo k, k is the order of (k) /2 irreducible polynomials in Fp [z] .

We can determine the order of reducible polynomials with the following two results.

5.9 THEOREM ([7]). Given imducible polynornials 41, E F, [x] with 41 ( 0 ) # O , & ( 0 ) # O, ord(&&) = l ~ m ( o r d ( + ~ ) , ord(&)) .

5.10 THEOREM ([7]). g 4 E Fp [XI is an imducible polynomial of degree m with 4(O) # 0 ,

then ord(@) = p b o r d ( ~ ) , where b is the smallest intqer with 3 m.

For our purposes, since p > 2 and m = 2, b will be 1 .The comection between the compan- ion matrices and the polynomials is direct.

5.1 1 LEMMA. I/O(z) be an imducible polynomial in FP [ X I with d(0) # O, then ord(&') ) =

ord(d).

5.12 COROLLARY ([7]). Let +(x) be an imducible polynomial in Fp [XI with +(O) # 0.

Then o r d ( ~ ( ~ ) ) = ord(+)l) = pbord(d(x)) where b is the srnallest integer with 2 m.

5.13 THEOREM. Let V be cyciic with respect to linear transformation A. Let $(z) be an

ineducible, monic poiynomial and the minimal polynomial of A. As a permutation of the

non-trivial elements of V . A is a prvduct of cycles of a length ord(6).

PROOF: Since V is cyclic with respect to A, then is some v E V such that m y w E V can

be written as w = + ( A ) v for some t/, E F[x] of degree l e s than or equal to the degree of

+(x). NOW, let ord(q5) = k . Hence 4 ( x ) 1 ( x k - l), and thus - I = O. For any w E V ,

(Ak - I)w = O ===+ A'W = w and w is in a cycle of length at most k.

CHAPTER 5. CAYLEY GRAPHS OVER Z x Zp WITH P PRIME. 47

Now say Amw = w for some integer m > O. We write w = d ( A ) v , and hence

.-lm$(A)v = $(A) v and ( A m - I )$(A)v = O. Any polynomial which annihilates v annihi- lates V and thus +(x) will divide it. Hence, 4 ( x ) l ( ( x m - l)+(z)). Since d(z) is irreducible

it must divide either ~ ( x ) or zm - 1. In the first case, $(x) = d(x) since the degree of +(z)

is at most the degree of Q ( x ) In this case w = O. Otherwise, Q(z) 1 (zm - 1) which implies

that rn 2 k by the definition of order of a polynomial. Thus m = k and d l elements in the

space are in cycles of length ord(d).

5.14 THEOREM. The cycle inder of GL(2, p) acting on Z, x Zp \ (0, O) LF

where Ot(d, e) = @ ( d ) @ ( e ) ifd # e und 0(d)2 - 1 othenuise.

PROOF: If we consider A E GL(n, p) as a permutation a A ( x ) , it is a product of disjoint

permutations aa, a*, . . . ar, where each Ai is a hypercompanion rnatrix. The vector space

V = Z, x Zp is divided into subspaces Wi such that Wi is cyclic with respect to Ai for

each i. An element x in Zp x Z, can be uniquely expressed as wl @ w2 @ . . . @ w, with

w; E Wi. If x E Wi for some i, then by Theormi 5.13, under a~ it is in a cycle of length ord(Ai ) . If it is the direct sum of non-trivial elements nom more than one subspace, say

bv;, W;, . . . , W;, it is in a cycle of length lcm(ord(A;), ord(A;), . . . , ord(A;)).

CHAPTER 5. CAYLEY G W H S OVER L p x Zp WTH P PRIME. 48

To determine the cycle index we sum over the different types of rational normal forms

as uncovered in the previous discussion. The h t summand, (5.2), gives us the cycle in-

ventory for elements with the rational normal fonn A = D [ x - a, (2)]. As noted earlier,

matrices of thïs form have the effect of scalar multiplication. Hence the cycle structure is straightforward. The number of transformations for each order is the number of elements

in Z, with order ord(a), or cb(ord(a)) by Lemma 2.7.

V is also cyclic with respect to transformations with ra t iod normal fonn D[t2 + ax + bo ( 1 )] for irreducible polynornials x2 + a x + b. Theorem 5.8 gives us the number of poly-

nomials of degree two with a given order, hence we have (5.3) to account for these types of

transformations.

The third summand, (5.4), accounts for the matrices with rational normal form A =

D [ x - a, (O, 1 )]. There are two cyclic subspaces, one with order ord(a) and a second of

order ord( ( x - a)2) = p ord ( a ) , by Theorem 5.1 O. Both subspaces have p elements, but the

remaining elements in the subspace also have order p ordf a), since that is the least common

mu1 tiple of the pair.

The last, and most complicated, type of rational normal form cornes fkom matrices with

the rernaining normal form, D [ z - a, ( l)] D[x - b, (l)] when a # 6. Here there are two

cyclic subspaces, one each of orders ord(a) and ord(b). Thus, the elements in each ofthese

respective subspaces are organized in cycles of lengths ord(a) and ord(b), respectively. The size of both subspaces is p. The remaining (p - 1)? elements in the subspace are in cycles

of length fcm(ord(a), ord(6)). This is also a consequence of Theorem 5.9. There are @ ( a )

elements of order a and @(b) elements of order 6. However, we do not include the case

when a = 6 since they are special and are covered in (5.4). This gives us (5.5).

*

5.15 COROLLARY. n e number of non-isomorphic Cayley digmphs on Z, x Z, for pprime

CHAPTER 5. CAYLEY GRAPHS OVER Zp x Zp WlTH P PRIME.

is

with W defied in Theorem 5.14.

EXAMPLE. The cycle index for G L(2 ,3 ) acting on Z3 x Z3 \ (0, O) is

1 - ( 1 2 ~ 2 ~ : + 1 2 ~ 4- 6x4 + 8z6z2 f S X ~ X ~ + 2: f z;). 38

Hence the number of non-isomorphic Cayley digraphs on Z3 x Z3 is

1 -(1.2.23-22 + 1 2 . 2 + 6 - 2 * + 8 - 2 - 2 + 8 - 2 2 . 2 2 + 2 4 + 2 8 ) = 18. 48

The generating function for valency is

1 + + 2 ~ 2 + 2 ~ 3 + 4 . 2 + 3 2 + 2~~ + U? + us-

To determine the analogous result for ordinary graphs requires a modification of the

action of GL(2, p) on Z, x Z,. As in the circulant case we are looking at the group of auto-

morphisms acting on the quotient group, in this case (Z, x Z, \ (0,O)) /{(1,1), (- 1, - 1 )}, which we shail denote 2. In the circulant case it simplified matters to pare down the auto-

morphism group. In this case, it is simpler to use GL(2, p).

We have already detennined the nature and size of the conjugacy classes. However, the

cycle structure of a transformation is different when we identie x and -z in ZP.

5.16 LEMMA. Conrider monic, imducible, 4 E F, [x] with k = ord(4). Let k' be the smallest integer such that +(x) divides xk' + 1, ifit 4xists. Then kf = k/2 i f k is even and

does not acists if k is odd.

CHAPTER 5. CAYLEY GRAPHS OVER Z p x Zp WTH P PRIME- 50

PROOF: If k = .Id, then +(z) divides z2d - 1 = (xd - l ) ( x d + 1). By the definition

of order, q5 must divide xd + 1. Thus k' 5 d = k / 2 . If k' < d, then 4(x) divides (zkr + 1 ) ( x k 1 - 1) = z~~~ - 1, contradicting the order of 4.

I f k is odd, then as in the previous case certaidy if k' existed it could be no l e s than k / 2 , no more than k and must divide k . Thus, it does not d s t .

This k' serves as a new form of order. If V is an n4imeasio-na1 subspace cyclic with respect to A E G L (n, p ) , then the length of a cycle of A acting on an elernent of V where x and -x are identified is k = ord( A) if k is even and k / 2 if k is odd. We capture this action with the notation

x z is odd

x / 2 otherwise.

5.17 THEOREM. The cycle indexof G L ( n , p ) actingon Z is

where a' is as defined in Theorem 5.14, and

[ l cm(d , e)/2 Borh d and e aie ewn and am divisble by the sarnepowers of two L(d , e ) =

( l m ( d , e ) orhenuise.

CHAPTER 5. CAYLEY GRAPHS OVER Z p x Zp WITH P PRIME.

Figure 5.1: THE FWE ISOMORPHISM CLASSES OF X(Z3 x S )

5.18 COROLLARY. The number of non-isomovphic Cayley gruphs on Zp x &for p > 2 prime is

where QT' is as defined in Theorem S. 14. and L is as defined in neorem 5.17

EXAMPLE. The cycle index for GL(2,3) acting on Z3 x Z3 \ (0,0)/{1, -1) is

1 -(12x21; + 12x4 + 61: + 16x3x1 + 22:) 48

CHAPTER 5. CAYLEY GRAPHS OVER Zp x Zp WTH P PRIME

hence the nurnber of non-isomorphic Cayley graphs on Z3 x Z3 is

as illustrated in Figure 5.3. The valency generating fiinction is 1 + u2 + u4 + u6 + us, also as expected.

5.4 Z; and Beyond

The methods we have dweloped so far this chapter have been sufficiently general that they can be used to count Cayley graphs over Z: and any other CI-Group of the form Z,". How- mer, a a better notation is essential. The following notation was deriveci by H. Fripertinger [3] for the cycle index of G L ( n , p) acting on Z:. It encapsulates much of the work and

removes the intuition, but allows one to calculate and so it is included.

5.5 The Cycle Index of GL(n, p)

To facilitate the description of the cycle index we introduce a product notation. Let A and

B be polynornials in indeterminates x 1, 2 2 , . . . such that

Defme the operator o on A and B as follows

where

CHAPTER 5. CAYLEY GRAPHS OVER Zp x Zp WITH P PRIME.

We will denote the kth pwer of this operator by A ( x l , . . . , zn)"*.

We have already determineci the cycle structure and size of a conjugacy class. The last task for determining the cycle index of GL(n, p ) is to determine ail possible normal foms.

Let p be the M6bius fiinction. There are

monic, irreducible polynomials o f degree d over Z,. Each monic, irreducible polynomial of

degree at most n witb the exception of $(x) = x can occur as a divisor of the characteristic

polynomial of a matrix A E G L ( n , p ) . Label these t , = CL, N,(i) - 1 polynomials

as 91 (2) , &(x), . . . , &, (z), Ulith the degree of di (=) as di- We need to h d al1 lutio ion^

where y; is a non-negative integer. For each solution 7 one must determine the possible

cycle types, that is, the partitions A(') of ri. Cal1 this set CT (7;). The representative of the

conjugacy class of matrices A with characteristic polynomial

We now have our cycle index for G L(n, p ) .

kd, - ( k - l ) d , w h e ~ e ÿ = ~ r d ( + i ( x ) ~ ) . Furthemore = P

ci k , [ p 1, is the onter of GL(n,p) ,

and b(dj, A( ' ) ) Ls the ske of the centralizer of D(4j , A(')) as computed in 5.4. Thefini sum

CHAPTER 5. CAYLEY GRAPHS OVER Zp x P r p WITH P PRIME. 54

runs over ail solutions y = (-yi, . . . , yt,) of 5.1 6 and the second nurr over all t,-hrples X = (ML), . . . , Mt,)) E nfzi CT(7;) .

The formula from Theorem 5.19 can be used directly to eaumerate digraphs in a manner simiia. to every other family of Cayley digraphs we have encountered so far. W1t.h sorne ad- ditional notation as one could also develop a formula for Cayley graphs. Lemma 5.16 gives

how to modiS, this cycle index to get the cycle index of GL(n, p) acting on Z;/(l,, -1,).

5.6 Final Thoughts

Cayley graphs and digraphs are a very elegant family of graphs. By demanding a specific re-

lationship between the automorphisms of the group and the isomorphisms of Cayley p p h s on the group, we have been able to develop a nice way to enurnerate the non-isomorphic

Cayley graphs of a given order for certain CI-groups. As more groups are identifieci as CI-groups, this method can be revisited as a method of enumeration. Since currently most

energy in the CI-group problem is directeci at products of cyclic groups, a better form for

2 ( G L (n, p) , Z,") would produce more sa t i smg mumeration results. For every case that

P6lya's Enurneration Theorem is used, it may be worthwhile to investigate other uses of the cycle index apari from straight enumeration. For example, one could cuunt the number

of distinct edge colourings using k colours by substituting k + 1 for each xi in the cycle

index. A different, though likely interesting, epilogue to these results h a e would be the

computation of asymptotics. Of particular interest may be to determine the percentage of

circulants which are unit circulants.

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