, .-- * - - - - , A K P S B I E DER WISSWSCHllFTEN DBR üDR

PREPRINT

Michail Ch. Klin and Reinhard Pöschel

The isomorphism problern for circulant digraphs

with pn vertices

(Comuiunicated by H. Koch)

Keywords

C i r cu l an t graph

Graph isomorphism

Ad6m's con jec tu re

Isomorphism problem

CI-graph

Schur r i n g (S-ring)

AMS Subject c l a s s i f i c a t i o n (1980)

05C20, 05C25, 20B25, 68E10

Received November 27th , 1980

ABSTRACT

A (directed) graph with vertex set z,={o,I, ... ,r-lf is called circulant over Zr if its automorphism group contains the cycle (0 1 ... r-1 ) . In this paper, a necessary and sufficient condition for circulant graphs over Zr (where r=pn, p being an odd prime number, n€lN) to be isomorphic is given. This result proves a generalization of ad&tnqs conjecture which

holds for n=l but fails for n) I. hloreover, an algorithm de- ciding whether ad&'~ conjecture holds for a circulant graph

over Z pn

(i.e. whether the graph is a CI-graph) is presented. The proof of the isomorphism theorem uses the method of

Schur rings (S-rings). In order to make the paper self-con- tained, the results on S-rings are developed (and proved) as far as necessary; however the results are of their own in-

terest, too.

Ein (gerichteter) Graph mit der Eckpunktmenge z,={o ,I ., ,r-13 heiBt zirkulant, wenn seine Automorphismengruppe den Zyklus 0 I . 1 ) enthslt . In der vorliegenden Arbeit wird eine notwendige und hinreichende Bedingung dafiir angegeben, daf3

n zwei zirkulante Graphen uber Zr (mit r=p , p ungerade Prim- zahl, n€BJ) isomorph sind. Das Ergebnis beweist eine Vesall- gemeinerung der Hypothese von adfun, die fur n=l ,nicht aber fiir n > I gilt. Dariiberhinaus wird ein Algorithmus angegeben,

der entscheidet, ob die adhsche Hypothese fur einen gegebe- nen Waphen iiber Z PE e: ilt (d.h. ob ein CI-Graph vorliegt ). Der Beweis des Isomorphietheorems verwendet die Methode der Schurschen Ringe (S-Ringe), Die Ergebnisse uber S-Ringe werden so weit wie natig entwickelt (und bewiesen); sie sind dariiberhinaus auch von eigepstandigem Interesse.

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CONTENTS

$ 1 In t roduct ion ......................................... 3 $2 Main r e s u l t .......................................,. 7 $3 Proof of t h e theorem (sufficiency 2.3(2)+[1)) ...... 10

54 S-rings and S-isomorphisms .......................... 1 2

$5 Graphs, groups and S-rings ..................... (restatement of t h e problem) 18

6 Proof o f t h e theorem (necessi ty 2.3C1)+(2)) ........ 22

................... $7 Cayley graphs and the CI-property 25 ....................................... 9 8 Some examples 31

REFERENCES ........................................... 38

$7 In t roduc t ion

The isomorphism problem f o r graphs c o n s i s t s i n f i n d i n g

llgoodtl necessary and s u f f i c i e n t c o n d i t i o n s f o r graphs t o be

isomorphic. From t h e t h e o r e t i c a l po in t of view t h e r e i s no-

t h i n g t o do: There always e x i s t s a f i n i t e a lgori thm t o de-

c i d e (e.g. by checking a l l b i j e c t i v e mappings between t h e

v e r t e x s e t s ) whether two f i n i t e graphs a r e isomorphic o r not.

But even a high-speed computer could not manage t h e number

of computations necessary f o r t h i s a lgor i thm i n genera l . One has t o r e s t r i c t t h e c l a s s of graphs under c o n s i d e r a t i o n f o r

more e f f e c t i v e r e s u l t s .

In t h i s paper we cons ide r d i r e c t e d c i r c u l a n t graphs. The

isomorphism problem f o r such c i r c u l a n t graphs i s of g r e a t

importance f o r a p p l i c a t i o n s ( f o r some more d e t a i l s c f .(A], [KIJP~.~). We completely so lve ( i n 52) t h e isomorphism prob-

lem f o r c i r c u l a n t d igraphs w i t h pn v e r t i c e s (p an odd

prime number). Moreover we mention an easy a lgor i thm deci -

d ing whether l l ~ d & n t s Conjecturet1 holds f o r a g iven c i r c u l a n t

d igraph (57) . Throughout t h e paper we w i l l use t h e word g r a ~ h --- - f o r

d i r e c t e d g r a p h s (d igraphs) without m u l t i p l e edges,

Therefore a graph i s simply a b ina ry (not n e c e s s a r i l y sym-

m e t r i c ) r e l a t i o n on t h e v e r t e x s e t !

f .f Def in i t ions . Let r (where denotes t h e s e t of - a l l n a t u r a l numbers) and Zr = {0,f , ... ,r-13. A graph

LfZrxZr wi th v e r t e x s e t Zr i s c a l l e d c -__---_-___-- i r c u 1 a n t ( o r

c y - ------- c 1 i c [ k l . / ~ b ] ) i f t h e r e e x i s t s a s e t I? c Zr - such t h a t

t h e r e i s an arrow from x t o y

( l e e . (x ,y)a m ) i f f y - xer ( a l l a r i t h m e t i c i s done modulo r ) , For s i m p l i c i t y and

without l o s s of g e n e r a l i t y we demand 0#I! f o r c i r c u l a n t

graphs. It i s easy t o see t h a t t h e c i r c u l a n c y of i s

c h a r a c t e r i z e d by t h e proper ty

( o r i n o t h e r words, 5 i s c i r c u l a n t i f f Autih c o n t a i n s t h e

c y c l i c group genera ted by t h e permutat ion (01 2.. .r-1 ) ) . We have a 1-1 correspondence between t h e c i r c u l a n t graphs

iti and t h e i r s e t s I! :

I? = { y r z r I ( o , Y ) ~

B = f (X,Y)EZ,XZ~ ( y - x ~ r j . Therefore i n t h e fo l lowing we s h a l l speak of t h e c i r c u l a n t

graph I? EZ, i n s t e a d of t h e corresponding 8 .

Examg&e. ---- I? = 1 5 Z i s t h e fo l lowing c i r c u l a n t graph a

over Z, :

We denote by

BUT I?

t h e ~ ~ - i ~ - ~ _ o _ r ~ - h I ~ s ~ g - z . 2 ~ ~ of 6 o r I!, resp . It con-

sists of a l l permutat ions f : Zr -+ Zr s a t i s f y i n g

(xf denotes t h e image of x by f ). l o r e o v e r , t h e c i r c u l a n t

graphs I? a r e c h a r a c t e r i z e d by t h e proper ty t h a t AUT r c o n t a i n s 2; - t h e r i g h t --- --I r e g u l a r -no ------ ---I r e ~ r e s e n t a - ----------- t i o n ------ of Zr c o n s i s t i n g of all permutat ions

a'': x c t x + a (acZr).

Two c i r cu l an t graphs I?, T c - Z, a r e i --------- s o m o r 2 ----- h i c

(no ta t ion F I? ' ) iff t h e r e i s a permutation f : Zr --+ Zr

such t h a t

Isomorphism c r i t e r i a f o r c i r cu l an t graphs were invest iga- t e d by many authors (c f . r e fe rences i n [K~/PO]). There i s a well-known c o n j ,,,- ,,,,,,,,,, e c t u r e o f ,-, A. ,-,,,--,,) Adam namely

T,I? ' 5 Zp a r e isomorphic i f f t h e r e e x i s t s an m

r e l a t i v e l y prime t o r such t h a t I? = r m (mod r ) . Unfortunately k d h ' s conjecture i s not t r u e i n general . It holds f o r squarefree numbers r = P ~ * . . . ~ P ~ a s shown by V.N. Egorov and A . I . Markov [ ~ g / ~ d . On the o the r hand B. Alspach and T.D. Parsons ([Al/~c$) proved t h a t r must be equal t o 2em where ee{0,1 ,2) and m = l o r m i s squarefree and odd i f Wd&mfs conjecture holds f o r a l l c i r cu l an t graphs over

zr . A d h l s conjecture remains v a l i d a l s o f o r some very spec i a l c l a s s e s of c i r c u l a n t graphs over Zr (r a r b i t r a r y ) inves t iga ted by D.Z. Djokovi6, B. Elspae, S . Tdida and J. Turner ([~jf , f~l/Tu],fTo], cf. [%d(p.191 )]).

But what can be done i n t he general case where r i s not squarefree? A s shown by t h e authors i n 1975 (cf . e,g.

[ P G / K ~ ( ~ . 5.1 9 ) I ) f o r r=p2 the re can e x i s t isomorphism c r i t e - r i a very similiar t o Adfun's conjecture. I n t h i s paper we genera l ize t h i s r e s u l t .

I n we formulate an isomorphism theorem f o r c i r c u l a n t

graphs over Z pn*

This theorem solves problem4 and 5 s t a t e d i n [Ad(pp. 191 ,I 93J.7 f o r graphs with pn ve r t i ce s . The proof e s s e n t i a l l y uses t h e method of Schur-rings (S-rings) , i n par- t i c u l a r t h e desc r ip t ion of a l l S-rings over fib;]. Nevertheless t he reader need not be fami Zfn i a r given with t h e in theory of S-rings; a l l needed p rope r t i e s a r e formulated and proved i n and without much S-ring theory i n o rder t o make t h e paper self-contained (of course, t h i s extends t h e

paper cons iderably) . A s a c o r o l l a r y t o our main theorem we

formulate i n a necessary and s u f f i c i e n t cond i t ion f o r a c i r c u l a n t graph t o s a t i s f y Ad&nfs con jec tu re . The proof of

t h i s r e s u l t w i l l be publ ished elsewhere,

The r e s u l t s a r e i l l u s t r a t e d by some examples g iven i n

t h a t The p resen t au thors t h i n k t h e methods used i n t h i s paper

can a l s o be a p p l i e d t o t h e genera l case y i e l d i n g an isomor-

phism theorem f o r a r b i t r a r y c i r c u l a n t graphs wi th r v e r t i c e s

- provided t h a t i t i s p o s s i b l e t o f i n d a c h a r a c t e r i z a t i o n of

a l l S-rings over 2,. Some i n v e s t i g a t i o n s i n t h i s d i r e c t i o n

a r e done r e c e n t l y by Ja.Ju. Goltfand (AN SSSR, Moscow).

Be Alspach and T.D. Parsons i n v e s t i g a t e d t h e isomorphism

problem wi th d i f f e r e n t methods. The i r r e s u l t f o r n = p2

can be found i n [~l/Pa(Thm.3, p.1 07)] and as Prof . T.D. Par-

sons poin ted o u t t o u s he h a s some r e s u l t s f o r r = p n ( n ) 2 ) , too. A s d i scussed i n [ ~ l / P a ( p . 107)] t h e r e i s now some hope

t o so lve completely t h e isomorphism problem f o r c i r c u l a n t

graphs.

There i s another problem c l o s e l y r e l a t e d t o t h e isomorphism problem, namely t h e so-ca l led KEnig-problem ( c f . e ,g. 0101, [ P ~ / K ~ J , [K1/~8]) : Which permutation groups occur as automor-

phism groups of c i r c u l a n t graphs wi th r v e r t i c e s ? V.A, Vy-

gensk i j , P.H. K l i n and N.I. Eerednizenko solved t h i s problem

comple te l~ r i n caae rzp3 (and n e a r l y f o r rSPn, t o o ) and worked

o u t a computer program f o r t h e c o n s t r u c t i o n o f a l l S-rings 3 over Zr ( r=p 1. These r e s u l t s provide (among o t h e r t h i n g s )

a c l a s s i f i c a t i o n and e f f e c t i v e l i s t i n g of a l l non-isomorphic

c i r c u l a n t graphs wi th g iven a u t omorphism group (, [Vy/~l/Ee]).

ACRNOVfLEDGEMENTS, We wish t o express our thanks t o P ro f ,

L.A. ~ a l u z n i n who had d i r e c t e d our a t t e n t i o n t o t h e i n v e s t i -

g a t i o n of Schur r i n g s and showed a cont inous i n t e r e s t on our work. O u r thanks a r e a l s o due t o P ro f . A. ad& f o r some sti- mulat ing d i s c u s s i o n s and remarks.

92 Main r e s u l t

Before s t a t i n g o w main r e s u l t we in t roduce some more no-

t a t i o n s used i n t h i s paper.

2.1 Notat ions. Let Sr ( r e m ) be t h e ful l symmetric group - of a l l permutat ions f : Zr--+ Zr. The a c t i o n of some f t S r

f on an element xeZ, i s denoted by x . The composition of

f , f f c S r i s def ined by

X f f l f f f :=(x ) .

For s u b s e t s G , G 1 S Sr and xeZr l e t

G G f :={ggl 1 gpG, g f r G f ] (and Dg:=GG1 f o r G1=lg3),

xG :=I@ lgeG].

G ~ : = { ~ E G I a k a 1 i s t h e s t a b i l i z e r --------------- of G a t t h e po in t

arZr (we always choose a=O). Dealing wi th Zr ( i n p a r t i c u l a r

r=pn) a l l a r i t h m e t i c ( a d d i t i o n , m u l t i p l i c a t i o n ) i s done mo-

dulo r ( i .e . i n t h e r i n g 2,). If t h e r e a r e diffentntmodules

under c o n s i d e r a t i o n we w r i t e e x p l i c i t e l y

x a y m o d r l

when x equa l s y modulo r f . For xcZr, rls r ,

[ ~ I m o d r (o r mod r f ) )

means t h e i n t e g e r which i s l e a s t h a n r f and congruent t o x

modulo r l .

For T,T l $ Zp, xrZr we s e t :

T + x :=[t+x 1 ~ C T ]

~ + ~ l : = { t + t l I t , t l c ~ f j

Tx :={tx I t r ~ ) and pf:={tf I tcrJ f o r f€sr.

The g r i m e r e s i d u e c l a s s ,----- ,-,,,,,--- ,-,,,,,- ~-cOXF! of a l l numbers

qcZr r e l a t i v e l y prime t o r i s denoted by

I P ( ~ ) : = [ ~ ~ z ~ ( g.c.d. ( q , r ) = 1 f .

Let r = pn (n E 3T). Then we def ine :

P; :={]pi I X=0,l, ...,p n-i-ll ( O s i d n ) .

mote t h a t n-1 i i+l

n-j )pJ = P(P"){P ,P 9 - 9P n-1) ~ f ; = u P ( P

j =i (mod pn).

2.2Notat ion. For T 5 Z n - we def ine P

n-i i r ( i ) := { X B T I g.c. d . (x ,pn)=pi ) =I! n ~ ( p )P . n-1

Clear ly , 3?= u 3?( i )* i = O

Now we a r e ready t o formulate the theorem ( the proof i s

given i n $3 and 56).

2.3 Main Theorem. I! , r ' 5 Z be (d i r ec t ed ) c i r cu l an t ===Zf======= P -

graphs, p 7 2 prime number, neBJ. -- Then t h e following condi-

t i o n s a r e equivalent: 7-

(1 ) I! - and I! ' - a r e isomorphic : r Z F. ( 2 ) There e x i s t mo,ml ,- ,mn-l r P ( p n ) -- such t h a t

y i ) = q i ) mi ( 0 5 i j n - 1 where - the following llequality conditions1' - a re s a t i s f i e d :

j -i ( l + p ) r ( i l # T ( i ) # % , 0 6 i < j S n - l , - - then

mod p j -i 1 mi i mi+l

mod p jmt ( i s t ~ j - 1 )

m j l z m j mod p .

2.4 Remarks. - ------- a ) The above "equa l i ty cond i t ions t t can be w r i t t e n a l s o i n

t h e form:

If P + ( ) # q i ) 7

f (8 ( O S i 4 j g n - 1 ) t h e n

(*I$ mi mt mod p jmt+' f o r i g t ( j . (This immediately fo l lows from Lemma 3.2, too ) .

b ) Because I.? = T(i)ml (mod pn) i f m l m l n-i (i Irn (modp ),

t h e m i l s i n t h e above c o n d i t i o n 2.3(2) can be chosen as

m E (0 6 i n-1 ); t h e r e f o r e e,g, we can t a k e i mi=mi+l =... =m i f (1 +p n-1 -i

n-1 ) ' ( i ) f ' ( i ) '

c ) If 2.3(2) i s f u l f i l l e d t h e concre te form of a n isomor-

phism (which has t o e x i s t ) can be found i n t h e next para-

graph $3.

d ) What concerns t h e case p=2 , i t i s not q u i t e c l e a r whether

theorem 2.3 remains v a l i d . The answer seems t o be "yesv.

However t h e proof depends on t h e d e s c r i p t i o n of S-rings

over Z2" which r e c e n t l y w a s g iven by Ja.Ju. Gol t fand,

M.lI. K l i n and N. Najmark (unpublished r e s u l t , Oct, 1980).

The isomorphism theorem f o r c i r c u l a n t graphs over z2"

i s no t y e t formulated but i t followa immediately from

t h i s d e s c r i p t i o n .

53 Proof of t h e theorem ( s u f f i c i e n c y 2.3(2)=>(1 ) )

The proof 2 .3 (2 )+(1) , namely t h a t t h e c o n d i t i o n s i n 2.3

a r e s u f f i c i e n t , i s more o r l e s s t e c h n i c a l l y . We need t h e

fo l lowing lemma:

3.1 Lemma. - I n Z we have P --

We omit t h e p r o o f o f t h i s well-known number t h e o r e t i c a l

r e s u l t (cf , e.g. Da($5)]). I

ITOW, l e t I?, F '=Zpn and mo,...,m n-1 be g iven such t h a t

2,3(2) i s f u l f i l l e d , shg.l_ p-o_pg ' by - c o n s t r u c t i n g -----------

For each ie{0 , l , ... ,n-2) w e d e f i n e ki t o be t h e g r e a t e s t

number such t h a t i 4 - ki 6 n-1 and k, -1

mod p .L mi E mi+l

k - -t mt E d m t + l

mod p 1

%.-I smki mod p 1

i n case mi o mi+l mod p o r i f i=n-1 we t a k e ki=i.

Obviously

0 j k o $ k1 5 ... 6kn-25kn-l=n-I .

3.2 Lemma, For 0 $ i j n-1 -- we have

n n ' ( i ) ='(i) +pki+l - and I'li) =I ' i i)+Pki+l ,

------- j P r o o f . By d e f i n i t i o n of ki and cond i t ion 2 , 3 ( r ) i we

have (1 +pki+l -' ) F(i) = Z(i ) , t h e r e f o r e a l s o T i i )= I l ( i )mi - - ki+l -i =( l+p . )T(i)mi = (l+p ki+I -i )rti), Elor t h e lemma fo l lows

from 3.1. 1

Every xcZ pn

has a unique r e p r e s e n t a t i o n i n t h e form

t n-1 X = X(0)+X(l )P+. . .+X(t)p + * a .+X (n-1 )P

where O g x C t ) $ p - l .

Now we d e f i n e a permutation f6Spn as fo l lows:

where

For x , y t l n l e t X-ys~(pn-i)pim Then pi d i v i d e s x-y P

and we have x ( ~ ) = Y ( ~ ) f o r t < i , i , e .

n-1 k i X-Y =& E x ( ~ ) - Y ( ~ ) )pt + 9 - ki+l

We g e t k 4

k +I -i Because ki kt and m i a t mod p t t ( i .e . mip =mtp t k . + l ) mod p 1

f o r i s t 6 k i , we have

t t x ( t ) m i P = x ( t ) m t p t k ( t ) m t b o d $i+l-t r[x(t)mt]modpkt+l-t

k*+f *pt 5 ztpt mod p 1 . Thus k 2

I

Consequently, f f +pn (x;Y) c qj) +=+ X -Y a miT(i) ki+l = rii),

(cf, 3,2), i.e., f is an isomorphism. This finishes the

proof of Theorem 2.3(2)=$(1). 1

$4 S-rings and S-isomorphisms

For the proof of theorem 2.3(1)+(2) we shall use the me-

thod of so-called Schur rings (S-rings). We do not go into

details here and mention some properties of S-isomorphisms

of "basic quantities" af S-rings only. For more information

the reader is referred to [wie], [PO/K~],[N~~, [MIPO].

Revertheless this paper is self-contained as much as possible;

therefore no knowledge on S-rings is needed, almost all re-

sults are proved completely (one exception is the characte-

rization of the basic quantities of S-rings over Z n given P

in n6]).

4.1 Definitions and Notations. Let r s m and let - n

be a d i s j o i n t union of subsets with the following pro-

perties:

(i) %={03 ,

(ii) -T~E{T, ,TI , ... ,T~) where -Ti: ={r-x 1 xt!FiS ( = I [-xlmod r I xq$) 9

(iii) For yeTk, the cardinality pi of ,j

Tin(y-T.) depends only on j , but J

not on the choice of y (Osi,j,kfn).

R e It is easy to see that (iii) is equivalent to

ciii) If xeTi, yeTj and x+ycTk then all ele-

ments of Tk appear in Ti+T with the j

k same multiplicity pi (i,e,, counting ,j

multiplicities, T +T is the union of sui- i j

table Tkls with corresponding multiplici- k ties pi $.

9

Por zeZr let T(,) be the uniquely determined Ti such

that .€Ti. The system

with the above properties (i)-(iii) is called the system of

all basic ------- quantities -------------- of an S-ring over Zr. For

--I---- over short we will speak of the S - r i n g S =<T (z)>zEZr ------

zr

Remark. ------ For readers who axe interested in S-ring theory we

mention that a S-ring as precisely defined e.g, in [wie], fpa/~a] is nothing else thm the 2-module generated by the Tits in the group ring(^(^,);+,.>.

Let S=(T(~))~~~ and S'-(T[~)>~~Z~ be S-rings over Zp r

and let f: Zr + Zr be a permutation. The f is called

S - i s o m o r ~ h i s m of S onto S' if an ------------ ------ (T(,) +xlf = T1 f ($1 + X

f for all x,z6Zr. In particular we have (T(,)) =T1 (zf), 0 f =0.

Now we c o l l e c t some p r o p e r t i e s o f S-r ings and S-isomor-

P r o p o s i t i o n ( [ ~ i e (~hm,23,9(a))]). Let S = ( T ( ~ ) ) ~ ~ ~ ~

be an S-r ing ove r Z2 ( ~ E R ) . Then -- T ( z ) q = T(zq)

f o r all q e l P ( r ) . -- (The p r o o f i s similiar t o t h e proof of 4.3, ~ f . [ W i e ( ~ . 5 9 ~ )

4.3 'Proposi t ion, Let f be an S-isomorphism of t h e S-ring - -- Then we have s =('(z))nezr

f o r zeZk, q e E ( r ) . -

P ------- r o o f . (The proof f o l l o w s t h e i d e a o f t h e proof of h i e

(Thm. 23.9)]): If t h e s ta tement h o l d s f o r q , q q e P ( r ) t h e n

a l s o f o r qq' s i n c e f f - ( ~ ( ~ q q ' ~ = (T(yq)q ' ) = ( T ( Z q ) ) 9 ' -

f 4.2 - -4.2 q q ' = ( ~ ( , ) l fqq ' Thus i t s u f f i a e s t o prove t h e p r o p o s i t i o n on ly f o r prime nun-

b e r s q € P ( r ) . Consider U = TI(^)+ ... I +-T(z),

coun t ing t h e

9 m u l t i p l i c i t i e s . The m u l t i p l i c i t y of every element xl+...+x

q € U is d i v i s i b l e by q except t h e elementrs x+.. .+x = xq

(which appear wLth a m u l t i p l i c i t y ~ l m o d q ) , i , e , , except t h e ?.

elements of t h e simple q u a n t i t y T(,)q. The same ho lds f o r

U 1 = T1 +...+ T ' (q t imes ) where T' q i s e x a c t l y t h e (zf 1 (af 1 ( z f )

s e t of t h o s e e lements i n U ' which do no t have a m u l t i p l i c i t y

d i v i s i b l e by q . S ince f i s an S-isomorphism (cf.4.1) we

f have U =(T(z)+...+T )f =T' +...+I1 ( 2 )

= U1. (zf) (zf)

By 4.l(iii)l, U is the union of simple quantities of S

(Remark: This implies also that T(,)q is a simple quantity

what finishes the proof of 4.2). Therefore U1 must be'the

union of the images of these quantities (of coursewith tor-

responding multiplicities),. This yields (T(,)~)'=T' q. 1 (z 1

4.,4 Now we corisider S-rings over Z n ('p > 2 prime, n 2 1, ) . - P For a simple quantity

(3 1 *

is called the trace -___-_- of T (.z

By a result o.f R, Poschel

( [ ~ b ] ) , every S-ring S =(T ( , I)ZeZpn is fully characterized

by its -__- S - sy _------ s t em (cf. e.g. [~1/~6(2.7)],fi6(4.11 )J)

CCS) 7 = (Ao,% ,-,A,-,; e(S))

where

is a subgroup of the (multiplicative) prime residue class

group P(~") and B(S), for short 0, is the equivalence

relation on {O ,l , ... ,n-1! defined bg. 0 0

(i,j]eQ :* T = !y' (pi) (pj)'

Each % has the form

where Wi is a subgroup of a cyclic subgroup Fn of P(~") of order p-1 satisfying the property wzwl mod p +w=wl

for all w,w1eWn (cf. 4.5). By 3.1 , ki is the greatest num-

ber such that ( I + ~ I ~ ~ - ~ ) T fT (Pi) (Pi) '

The S-systems of S-rings were described e x p l i c i t e l y i n (PO].

We mention here only some p rope r t i e s needed l a t e r ( f o r proofs

see (Pal) . If [ : = I f i , j )€@j) , then (obviously)

T P and 1 %bpi f o r b € ~ ( ~ " ) . (pi (bpi )=

If ( i , j and i # j then [I]@ i a a f u l l i n t e r v a l 1 of t h e

0 u P ( P n-j j

F(pJ ) = T c P j ) = )p . E Li3@

1f [Ole={Oj , A,= wo+p; +I ('d.(p~)) a n d i f k o l l - then 0 n

[1Ie = 11) and A1 = Wo+ P" (mod pn-I 1, i.e. T. 1 =Wop+Pk +l , kl (P 1 1

with kl _Z ko (c f . [~0(4.11 1 1 ) ; i n case ko=O no condit ion

f o r A, i s required.

4.5 We give here some remarks concerning t h e s t r u c t u r e of - t h e prime res idue c l a s s group p ( p n ) (w. r . t . mu l t i p l i ca t i on

mod $). For more d e t a i l s see e.g. [~a].

Let w O ~ I P ( p n ) be a pr imi t ive roo t modulo p ( i .e . ,

~ ~ - ~ } z P ( p ) m o d p ) s u c h t h a t wo n 41 ,w0 ,w0 9 - 9 (, p-lsl mod p . Then every element x s B ( p n ) has a unique represen ta t ion

at' x = wo ( I +p )*I1 mod pn

(where oc' mod p-1 and 08 mod pn-' ). Moreover, every (multi-

p l i c a t i v e ) subgroup A of ( the c y c l i c group) 3P(pn) can be

(uniquely) writ-bsn I n t h e form

A = V + P: = W(l + P:) mod pn, 2 where W i s a subgroup of Wn:={l ,wo,wo,,,wo P-*!. W

4.6 Lemma. Let f be an S-isomorphism of 3 = <T > - - -- - 0 0

( 2 ) z ~ Z n onto S' = (T*{. . P - Then (T (P i 1 If = T i p i ) for 0 S i 6 n - 1 . - -

P

(4.7)

Remark. ...----- 4,6 Impl ies 8 ( S ) = 8 ( S 1 ) ( c f . 4.4).

0 - 1 0 ' P r o o f . .----.- Let ~ = { l I ~ l e ~ ~ ~ ~ ) f a n d J 1 = f 1 I p ~ T i f .

((P ) ) t

By 4.4 we know t h a t s t h e r e a r e j , j , s , s 1 such t h a t

j = [ j , j + l ,-., j + s j , J 1 = { j l , j l + l ,..., j l + s l ) * Therefore

0 These numbers must co inc ide because ( ~ ( ~ i ) ) ~ = g 1

( ( P by 4,3. It i s easy t o s e e $hat t h i s i m p l i e s j=j l and s=sl,

0 0 i .e . , J P J f . Consequently T' = T ' s i n c e irJ=J l . I

U P ) 1 (Pi)

4.7 Propos i t ion . Let f be a s i n 4.6. Then - - ---- - f o r a l l zrZ n . *(z) = T i = ) - - P

P ------- r o o f . By 4.4 i t s u f f i c e s t o show t h a t E(S)=(Ao, ... ,Anwl ; - €3). and c ( S ~ ) = ( A A , , . , A I ; ~ ~ ; ~ ' ) - a r e equal. By 4.6 we have

8 = 8 l . Take ~ C { O ,I , ... ,n-13 and l e t ( i , j )€0=0 l f o r some

n j f i . Then Ai= P ( p )= A: by 4.4.In case [i]e={ij l e t a€%,

l e e . T i a - (P )

- T(pi) . Thus we have ( ~ ( ~ i ) ) ~ a ~ . ~ ( T ( ~ i ) a ) ~ =

Z ( T ( ~ ~ ) ) ~ = T1 i f * Because (p i f o ) €!Pipi) by 4.6, t h e r e i s ( ( P 1

i a q ~ ~ ~ ( p n ) such t h a t (pi)f = qp . Consequently,

T 1 i q = T ' i f ' T 1 i f a = T 1 i aq ( ( P I 1 ( ( P I ) (P

=+ T i p i l a = T 1 i i.e. a t A i . ( P 1' Thus % c , A i . Analogously li.is%, i . e . 4-A+ I

As a statement on S-rings the proposition 4.7 looks like

follows :

4,8 Theorem. S-isomorphic S-rings over Z are equal. I - pn -

$5 Graphs, groups and S-rings

(restatement of the problem)

5.1 Let I ? , I ? * c Z pn be isomorphic circulant graphs,

f: 2i'p-Z pn

an isomorphism of F onto T * and

G=AUICI1, Gf =AUTrl. Because of circulancy the permutations

a : xc,x+a (aaZ ,) P

belong to G and G t (cf. 1.1) and we can assume of = O f without loss of generality (otherwise t a k e f(-0 )* instead

of f ) . Obviously we have G '= fol Gf , in particular -1 f G b = f Gof (since 0 =0, notations cf. 2.1,). Now we consider

the orbits of Go and G b which we denote by

T(Z):= z Go and TiZ):= z G b (Z~Z n), P and let'

A famous result of I. Schur [~ch] yields:

5.2 Theorem. S(G,Z n) (& consequently S(Gf,Zpn)) & - - P - an S-rins over - Z ~ n fl ([sch])

~ r o ~ o s i t i o n ( [ ~ l / P o (4.6)]). - Let f --- be as i n u. - Then

f -- i s an S-isomorphism - of S(G.2 n ) onto S(G1,Zp,). There- P - -

f o r e S(G,Zpn) =S(G1,Zpn) - - (by 4.8).

P -----.- r o o f (we fo l low t h e proof of 4.6 i n [ K ~ / P U ) , We have

(T(z)+~) f=(zGo+x)f= Gox*f =

because

f f u = f - l ~ ~ X ~ ( - ( x ) ) ~ c f - ' ~ x * f ( - ( x = )ln= f = f - ' ~ f ( - ( x ) ) * = G 1 (no te Z> 5 G n G 1 ) and

-1 oU=(of +xjf-xf = O , i . e . U ~ G ; .

f But (T.(,)+x) 5 T(.f) +xf i m p l i e s e q u a l i t y because t h e

c a r d i n a l i t i e s must co inc ide (no te U = G ; f o r x=O, i.e.

f ( T ( Z ) ) = T i f ) , i . e . f i s an S-isomorphism (c f . 4.1) and

z 1 we a r e done, I

The fo l lowing r e s u l t i s a c o r o l l a r y t o 5.3.

5.4 Corol la ry , Isomorphic c i r c u l a n t graphs over - z ~ " have t h e same automorphism group. --- (We r e f e r t o f ~ 1 / ~ 6 ( 4 . 6 ) ] f o r t h e p r o o f . ) I

Now we want t o r e s t a t e our o r i g i n a l isoniorphism problem

f o r c i r c u l a n t graphs, A t first we observe:

t h e union of c e r t a i n o r b i t s of Go = (AUT 11 )0: - - -

k Moreover, ( l+p ) I ? ( i ) f T(i) impl ie s (1 +P k ) T ( p i ) f T ( p i )

( i o , , . 1 , kc41 ,2,...,n-1j 1.

P ------- r o o f . Because x=x-0 I r impl ie s xg=xg-Ogc F ( f o r @Go),

k r is t h e union of o r b i t s o f Go. Now, l e t (1+p ) T ( i ) # T ( i ) .

If ( i , j ) c e ( S ) (s:=s(G,Z,n)) f o r some j# i then IP(p n - i I p i =

k = ' ( i ) by 4.4 and t h u s ( l+p ) l ? ( i ) = I ? ( i ) . Thus r i ]e ts)=i i i .

But t h e n T ( i )

has t o be t h e union of o r b i t s of t h e form

n-i k T ( p i ) q , q s P ( p 1, and (I+p ) T ( p i ) = T ( p i ) wouldimply k

('+P I r ( i ) = r ( i ) . k Therefore (1+p )T(p i )# T(p i )* 1

5.6 Propos i t ion . Consider t h e assumption - - (A), : Every S-r ing S =<T( ) over Zpn has t h e f o l - Z ) zeZ n -

P lowing proper ty :

If f : Zpn -+ Zpn i s an 3-isomorphism of S onto S - - - t h a n t h e r e e x i s t mo , ,. ,mn-l E E (pn) such t h a t - --

( T ( ~ i ) lf = ~ ( ~ i ) m ~ j -i -- t hen mi,, ,m s a t i s f y a n d i f ( l+p ) T ( p i ) # T ( p i ) j

t h e cond i t ions - mod p j-i

mod p j -t mt *t+1 mod p (Of i d % < j 4 n - 1 ) I .

Under t h i s assumption (A) , c o n d i t i o n 2.3(1) impl ie s 2.3(2) -- (cf. Main Theorem p . 8 I.

P r o ------- o f . Let f be an isomorphism between t h e c i r c u l a n t

graph I' and T 1 c Z n (i .e. 2.3(1) is f u l f i l l e d ) , o f = O P

(c f . 5.1 ). By 5.3, f i s an S-isomorphism of S = - S (AUT I', Zpn)

onto S1=S(AUTI",Zpn) - and we have S = S t (cf . 4.8). Under n t h e assumption (A) o f 5.6 t h e r e e x i s t mo,,.,mn,l~IP(p )

such that (T(p i ) l f = T ' and mi,...,mj s a t i s f y (*)$ . . (P i if ( l , + ~ ~ - ' ) ~ ( ~ i ) f . T ( p i ) .

If I ' ( i )=@ o r r ( i ) = P ( p n-i ),pi t h e n I? t i )= (d o r T1(i) =

= WP n-i)pi, i . e . , "(i)¶ '(i) mie

Otherwise [i]8(s)=jij and F ( i ) = u qP ilq (cf. 5.5) where seQ

Q:={~ s P ( ~ " ) I qpic T Therefore we have again:

(note that t h e f irst e q u a l i t y fo l lows from 4.6 arii t h e d e f i -

n i t i o n of an isomorphism). Moreover, ( l+pjoi) I! (i) $ J? (i)

impl ie s (r)! f o r mi,, ,m because o f 5.5 and assumption j

(A). Thus 2.362) i s f u l f i l l e d , too . B

$6 Proof of t he theorem ( n e c e ~ s i t p 2.3(1%)+('2))

With t h e r e s u l t s of t h e preceeding pmagraphs we a re ready

t o pTove 2,3(1)53(2), For t h i s reason we a r e going t o pTove

assumption ( A ) i n 5.6:

Let. f be an S-isomorphism 05 S = < T ( ~ ) ) ~ ~ ~ onto it- P

s e l f . We prove the exis tence of mop, ,mn-l 6 IP ($1 by in -

duction on nc Be

For n=l. WE have (T .4po) ) f= T ( l f ) = T(po)mo f o r

f mo:=(po)f= 1 (cf . 4.1, 4.2).

Remark. -I---- The resul-b f o r n=2 i s proved i n [~i5/~a(8 .5 , lgV.

Now, l e t 5.6(A)l be f u l f i l l e d f o r a l l n 1 < n and oonsidelp

t he above f and S. Let io be t h e g r e a t e s t i n t ege r such

t h a t ( o , i 0 ) ~ e ( s ) . By 4.6, ( Y ) f= 8 , i.e. f mapms (Pi01 (PiO) -

8 = I P ( ~ ~ - ~ ) ~ ~ onto i t s e l f ; the re fore (pi,) i=o

o n ($n\T i and we can conaider t he r e -

(P 0 ) n n s t r i c t i o n of f to Pi +,. Dividing all elements of Pio+l

0 by pie+' we ge t an S-ring

oves 2 no (where no - ( 1 ) ) defined by P

3 (ZpiO+l ) Moreover,

is an S-isomorphism of onto i t s e l f . By induction the re - -. e x i s t mo , - ,mn -1 e IP(pno) such t h a t

0 T -

( T ( p j ) ) = T b j ) m j (J = O , I ,Ow, n o -1)

0

whereby (1 +pJ- i )~(p i ) f 5 ( p i ) h p l i e s (*Ii f o r Gi, ... ,mi. 0 I I

Define mi :=mop - , mio+l +j :=mj, .... , mnO7 :=mn 0 0

-1 '

Since ? ( p i p io+l = T (pi+io+ll by d e f i n i t i o n , we have 1

(Bipi))f = T (P i xn i f o r ie{iO+ll,... ,n-1{;

moreover, ( l+pJ- i )~(p i )# T (p i ) implies (*)! f o r mi,- ,m j

(io+l s i < j s n - 1 ) .

What happens f o r i $ i o ? - We dis t inguish two cases:

Case --_--- 1: i o ) O . Then, by 4.4,

T'(pO)= ". = T(p i ) - - 0 P ( p n-i i )P , ( is i o ) . id

Therefore t h e mi (i f i,) can be chosen a r b i t r a r i l y (e.g.

I o r m =m i i,+1 ) because they always s a t i s f y t he condi-

Case ------ 2: io= 0 . F i r s t l y , l e t (1 +p )T (PO )=T (po l* 2hen we have t o f i n d an

mo e z ( p n ) with ( T ( ~ ~ ) ) ~ = T ( ~ o ) ~ ~ (and with no o ther condi-

t i o n ) . Take mo:=lf; *hen

BOW, l e t ( l + p k o ) ~ ( p o ) f T( 0 f o r some k o b 1 and choose P 1

ko a s g rea t as possible. Then again by 4.4 (c f . p.16)

24 ( 9 6 )

T ( p ~ ) = WQ+P: 0 +I TCpl) ' wop+p; +I 1 wi th kl - 2 k,. c l e a r l y (1 +pkl -I ) T ( ~ T

(P (o therwise

n T(P)=

=T +pn =Wop+Pk , c f . 3.1 ) , t h e r e f o r e (1 +pko-' ( P I kl 1 T ( P )

and ('as a l r e a d y proved above) ( t):~ is s a t i s f i e d f o r t h e

above cons t ruc ted ml , ... ,mk . It remains t o f i n d an mo c I P ( ~ " ) 0

s a t i e f y i n g (*)iO f o r mo, - ,mk , i .e . s a t i s f y i n g m o a ml 0

mod pkO and (r ):o f o r ml , ... ,mk . Thus t h e proof were f i n i - 0

shed i f t h e r e would e x i s t an ~ * E P ( ~ " ) such t h a t k and mo 2 ml mad p 0 .

To see t h i s , cons ide r ' (p)C T ( ~ ) .+ lp( l ) (p-I ). his impl ies

Since pmleT(p)ml and i . T ( l ) ) f o r p = l l f e ~ ( p n )

WR g e t

pal = xf + Y ~ ( P - 1 n f o r some x , y ~ T ( ~ ) = Wo+Pk Consequently, x p s yy mod p 0 + x s y mod p =+ x s y mod p kO+l . Thus pml t ypp mod p ko+l

* m1 zYp mod pkO.

Take mo:= y p . Then mo f u l f i l l e s t h e needed cond i t ions

because

(note T,(I )=T(y l ) and m o ~ m l mod p ko . Summarizing t h e r e s u l t s , we have found mo, ... ,mn-l

s a t i s f y i n g t h e c o n d i t i o n s i n 5,6(A). I

47 Cayley graphs and t h e CI-property

A s mentioned i n t h e i n t r o d u c t i o n t h e b e s t p o s s i b l e solu-

t i o n of t h e isomorphism problem f o r c i r c u l a n t graphs were

i d k n ' s con jec tu re , I n t h i s paragraph we m e going t o chmac-

t e r i z e e x p l i c i t l y a l l c i r c u l a n t graphs over ipn which s a t i s -

f y Ad&nls c o n j e c t u r e , i . e . , which have t h e so-ca l led Cayley-

Isomorphism Properky (GI-property). The r e s u l t i s an easy

consequence of t h e main theorem 2.3 however i t needs some

t e d i o u s cons ide ra t ions and %herefore t h e proof w i l l be pub-

l i s h e d elsewhere.

F i r s t of a l l we r e c a l l some no t ions .

7.1 A g r a p h i i ~ % , x B ( r a m ) i s c a l l e d a C a y l e y - r --- --- - g_r_-p-h of Zp if c A u t B . Thus t h e Cayley graphs of

Zr a r e e x a c t l y t h e c i r c u l a n t graphs over 2 . 8 i s c a l l e d

a C I ---- - g --- r aq-h o --- f Zp i f a Cayley graph 6' of ( t h e addi-

t i v e c y c l i c group) Z i s isomorphic t o a i f f t h e r e e x i s t s r an mcAut Zr w i th L 1 = p. Because Aut 2, = P ( ~ " ) , a &-

c u l e n t g r a ~ h T f Zr is a CI-graph i f f f o r e v e q c i r c u l a n t

graph T'g ZL- isomorphic 42? P t h e r e e x i s t s rncIP(pn)

such t h a t T' =Tm. That means, CI-graphs of Zr a r e those -- f o r which A d h ' s con jec tu re i s f u l f i l l e d .

'r i s a C I - g r o u g ---- ----- f o r ---- - g r a p h s --- ---- ( o r ~ = G I g r o u ~ ) ----- i f a l l Cayley graphs of Zr a r e CI-graphs; e.g.

'P i s a 9 - C I group but

Z~ n (n 2 - 2 ) i s not. For some more d e t a i l s

s e e [~l/Pw,[~a].

The p roper ty of being a CI-graph depends on ly on t h e automor-

phism group of t h e graph (cf. e ~ g . [B~],[K~/P~]I . Therefore

a permutat ion group 05 Sr i s c a l l e d an &$.._m_ gr -o -gz

(over 2,) i f G i s t h e automorphism group o f a CI-graph

of Zro Then ? i d b l s con jec tu re i s f u l f i l l e d f o r all c i r c u -

l a n t graphs wi th automorphism group G.

Using a r e s u l t g iven i n [~ l /P t i (~hm, 4.911 t o g e t h e r with

4,8 we g e t %he fo l lowing c h a r a c t e r i z a t i o n of ?id& groups

over Z . pn

7,2 Theorem. Le-k G be an automorphism group of a c i r cu - - - -- - - - l a n t graph over Zr where r = p n , p > 2 prime, n s N . - Then

C i s an Ad& ~roup i f f --7

[N (G) : O ] = [N (ZG) : N~(z~)]. 'r 'r

Hereby N ~ ( H ) = I @ ; C L 1 a= I&) denotes t h e n o m a l i e e r of H 4 - Sr

i n L b S r . B ( [ K ~ / P G ( ~ * Y ) ] ) *

For p r a c t i c a l a p p l i c a t i o n s t h e r e immediately a r i s e s t h e

question: How one can recognize a c i r c u l a n t graph I? p Zr

t o be a CI-graph without u s i n g t h e automorphism group? How

In avoid t h e g r e a t amount of cornputationa (e.g. on a compu-

t e r ) necessary f o r determining t h e CI-property v i a automor-

phism groups? The next theorem provides t h e answer f o r c i r -

cu lan t graphs over Zpn

Some pre l iminary c o n s i d e r a t i o n s a r e necessary:

7.3 Let T = U T(l ) U ... U ) 5 Z be a c i r c u l a n t - P

graph. Then t h e s t a b i l i z e r

Stab := { q c W p n ) 1 I?(i)q = I?(i) 3 i s a subgroup o f and t h e r e f o r e ( c f . 4.5) of the form

n Stab I?(i) = Wi +Ps (i = O , l ,... ,n-1 ).

L e t Wi be genera ted by di (cf . 4.5) and l e t s = ki+l -i

( thus i L - k i c .I n-I ). For T(i)= @ WB d e f i n e Stab T(i)= P (~") .

The n- tupels

a r e c a l l e d t h e c h a r a c - t i e r i z i n ~ ------------------- n u m b e r s y s t em --------- - ------- f o r I?. We reduce t h i e system ( k , f ) i n two s t e p s : ---- W

Step 1. : If g.c.d(Ii,p-l ) = I and k i t h e n d e l e t e t h e

ith compononent i n both , & and L, y i e l d i n g a sys-

t e m , say (k',hl) ( 0 5 i ~ n - I ) .

Step 2: If O ~ i l < i 2 6 k . & k il

then de le te t h e ipth com- =2

ponent i n and 1' tand s u b s t i t u t e Si by

The r e s u l t i n g system (kr t ,P t f ) W i s c a l l e d t h e F e d U C e d ---------- r e d r e d c h a r a c t e r i a i n g number system f o r T an denoted by (k W ,f ).

7.4 Theorem ( C h a r a c t e r i z a t i o n Theorem f o r CI-gra~hs] . -------------------------------- -- -- Let. I? g Zpn ( p > 2, n E B ) be a c i r c u l a n t graph o v e r - -- - Z ~ n

red- a n d l e t k - . j -- - , fred=(fj, : , ... ,f ) --- be i ts r e - s

duced charac ter iz ing- number system, Then I' is 3 CI-graph - - ( i .e , ad6m1s con jec tu re holds f o r I?) -- i f and on ly i f t h e -- fo l lowing - two -- c o n d i t i o n s - a r e s a t i s f i e d :

( i ) For a l l i { j 1 , except a t most one we have i = ki ----- ( i .e . , f o r a t most one index ie{j l ,... , jsi we have

(1 +PI r ( i ) # r ( i ) I *

( t i ) For a l l i , ,...,js}, i # i l , we have 1 -- g . ~ . d . ( ~ , ~ , ) = I ( e WiuWil genera tes t h e

whole group P ( ~ " ) f o r a l l d i s t i n c t i , i l e { j l ,...,j$).

Remark. ------ The empty system - kred=@, Ired=$ i s thought t o

s a t i s f y c o n d i t i o n s (i) and ( i i ) t r i v i a l l y .

The p r o o f of 7.4 w i l l be publ ished elsewhere. (1)

7.5 Note t h a t cond i t ion (i) can be checked wi th computing

t h e system k reduced by Step 2 on ly (without us ing any i i ) .

Therefore, i n many c a s e s , one can e s t a b l i s h t h a t I' i s - not

a CI-graph by computing - k red only. This procedure reduces

cons iderably t h e number of computations concerned. ( c f . 8.2)

Now, we g ive an a lgor i thm ( i n informal way!) f o x dec id ing

t h e CI-property f o r c i r c u l a n t graphs over Zpn (of course ,

t h e r e a r e many ways for f u r t h e r improvements of t h i s algori-t;hm),

7.6 Aigorithm dec id ina whether a c i r c u l a n t graph F 5 2 - - pn ( p 7 2 prime, n ~ m ) - - i s a CI-araph:

S t e p ( 1 ) : Working i n Z n , f i r s t l y f i n d a p r i m i t i v e r o o t P

wo modulo p s a t i s f y i n g wo ii 1 mod pn. (This can

be done as fol lows: Find a p r i m i t i v e r o o t U mo-

du10 p, i . e . , e.g., t a k e t h e l e a s t u.r lP(p) such

t h a t {u,u2 ,..., U P - ~ ~ Z [ I ,2 ,..., p-l] rnod p. Define

wo:= U (Pn-' mod pn. )

~ t e p ( 2 ) : Let a c i r c u l a n t graph F 5 Z pn

be given.

Det ermine n F(i),:= { X E T 1 g + ~ . d a (x,p ) = pi ) (0 5 i Sn-1 ).

~ t e ~ ( 3 ) : l o r a l l 0 1 , - 1 determine t h e l e a s t na-

t u r a l number ki such t h a t i f k i $ n - 1 and

X + pki+l (mod f o r a l l xei? ( i )

(ki is a l s o t h e g r e a t e s t nwnber such t h a t t h e r e i s

an ~ e i ? ( ~ ) wi th ~ + p ~ ~ + r ( ~ ) ) . i n c a s e I 4 t a k e &-g-/ = Lucckj t ; l ß ; - , $ , Let - k be t h e fo l lowing matrix:

s t e p ( 4 ) : For a l l i 0 1 , - 1 determine t h e l e a s t d i -

v i s o r fi of' p-1 such t h a t

W o Fixer(il (mod f o r a l i X E F ( ~ ) l

n (Remark: ti always e x i s t s inoa wo p - l x i x rnod p ).

s tep( .5) : Reduce - k and I as fo l lows: For a l l

i€{0,1~... ,n-lf, hf k i and - 1 i- &hen d e l e t e

(id !!i in k and I , r e s p e c t i v e l y . Denote - t h e r e s u l t of t h i s r educ t ion again wi th - k and

1, resp . We obtiain

If k 4 , t h e n I? i s a GI-graph, - - - - 0

Stcep(6): Reduce & and as fol lows: If (ii) and @j) are columns o f - k s a t i s f y i n g i < j $ k . $ki

3 t h e n d e l e t e i n k and f i n and

s u b s t i t u t e f i by t h e number l*c*rn . ( f i , f j )*

Do t h i s u n t i l 1 no such i , j e x i s t . One o b t a i n s

t h e reduced c h a r a c t e r i z i n g number system f o r I!

denoted by

r e d If k 4 t h e n I! is a CI-maph. - - - 0

If s = 1 t hen P i s s CI-graph. - 7 -- S t e p ( 7 ) : If t h e r e a r e a t l e a s t two colums ( ) [

kred -- such t h a t i#si - and j#kj t h e n

S t e p ( 8 ) : If t h e r e -- a r e two ( d i s t i n c t ) components Ii of 1~ - Fed -- such t h a t g . c . d . ( f i 9 f j ) # l ~ - t hen

F i s no t a CI-graph. ---

Step(9) r If (7) - and (8) -- are not fulfilled (i.e., if i=ki

for all but possibly one columns of k - red and

if g.c.d.(fi,i.)=I for all components of red) J

then r is a CI-graph. - 7 -

$8 Some examples

In this paragraph we mainly give some examples of iso-

morphic circulant graphs over Zpn. The isomorphism can be

easily established by theorem 2.3 whereas a direct checking

(e,g. in a geometrical representation or with adjacency ma-

trices) seems to be a hopeless task.

Let r = {I ,6,i i ,1)6,21 , 5 1

Then

The circulant graphs I? and over Z25 are isomorphic

by 2.3 (p0=2, rn1=3). According to the construction in $3,

the permutation

is an isomorphism from I? onto TI.

l? ia not a CI-graph because mo and mi cannot be chosen

to be equal. The graphs S! and ape represented in fi~,f

a d fig.2a. Here the arrow --> between subgraphs means

that there is a (directed) edge from each vertex of the first

subgraph to each vertex of the second one.

S! a d are isomorphic; here this can be seen alsa by re-

drawing the graph Ft as given in fig.2b. Comparing fig.1

and fig.2b it is obvious that the permutation 2 f t : x+yp 2x+3yp (modp )

is also an isomorphism from I? onto .

Fig. 2a

rf :

Fig. 2b

F i n a l l y , l e t us s e e how, t h e a l g o r i t h 7;.6 works when

app l i ed t o I? :

1 : W o = 7

0 1 step(3)l: & = t o

~ t e ~ ( 4 ) : f = (4.4)

S t e p s ( 5 ) , ( 6 ) : f e d - O '1, red = (4,4) - (u 1

S$sp(8) : i s not a CI-graph. ---

Then

Theorem 2.3 shows t h a t I? i s ieomorphic t o r1 but no t

t o I?11. More g e n e r a l , a l l graphs l? ' isomorphic - t o - a r e

e x a c t l y of t h e form

r = Pii) , r t i ) = r(i)mi (i=O,1 ,2 ,3) i = O

where mo 5 ml m2 5 m3 mod p .

From t'his also follows that I? is not CI. The algorithm 7.6

gives at once t'he Same result:

St'ep (3) : 01 23) 1 2 3 3

Step(7) : - - W is not a CI-graph.

Leti I' ={I ,28955, 9,36,63, 27,541

r'={113,40,67, 9,36,63, 27,541

rt=E 3,40,67, i 8,45 ,P, 27-, 545

Then 3 2 r(,)={i ,28,55{= I +P:= (I+P )r(0)+(i+~ )r(0)

(+ ko=2, cf.7.3)

"(1 )= @ (3 kl=2) b=ruai(4,bs3

2 4 1?(~)={9,36,63J= P + p3=(1+~)r(2) (+ k2=2) 4 ~(~)={27,54$= p3 (k3=3)

and

Tio)= -1 3 r(0)-13

Ti1 )= $ ='(I )=

r(2) ~ ( ~ ~ - 2

ris)= T(3) ?3)= r(3)

According to theorem 2.3, F is isomorphic .t;o T but not

to Tl1 . In general, al1 graphs I" isomorphic to are exac%- - - urf urf ly of the form = (, ( 2 ) (3), '(iIrni

mhere mo 5 m2 mod p .

From t h i s condit ion e a s i l y fol lows t h a t Adbrs conjecture

holds f o r I!. The algorithm 7.6 g ives t h e Same r e s u l t a s

f ollows :

Step(1) : wo.-I mod p n

Step63) : k = ( 2 1 2 3

Step(4) : = (2,1,2,1) 1- s t ep (6 ) : - k = (2 2) , f = (2.2)

So we have , f o r ins tance , I!' = T- 1 3 f o r t h e isomorphic graphs

I! and I!'.

We want to i l l u s t r a t e now t h e complexity of t h e c i r c u l a n t

graphs wider considerat ion. Therefore we present t he above

graph I? (see f i g . 3 ) using t h e following abbreviat ions:

/a\ denotes t he graph

a+54

> denotes t he graph cons i s t ing of two sub-

graphs /a\ and such t h a t t he re

i s a l s o an arrow from each ver tex of

/a\ t o each ve r t ex of .

denotes t h e graph A

%& A-A

denotes the graph

A -A !%A :+a d & A -A

Then F can be presented in the following way:

Fig. 3:

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INDEX OF IJOTATIONS

Authorsf addresses

IN 3

%r 3 r 3

AUT I? 4

z; 4 a" 4 , 1 8

X f 7

f f f 7

X G 7,

7 Ga - - - 7

[~lrnod r 7

4

D r . M. Ch. Kl in D r . R. Poschel

SU - 248025 Kaluga Akademie der Wissenschaften der DDR

u l . Valentiny N i k i t o j 39/44 Z e n t r a l i n s t i t u t f u r Mathematik und Mechanik

DDR - 1080 B e r l i n Mohrenstr. 39

- - -- -__- -~-

-. - - - -

A g 521 /3 ? S / 0 , v 3 f W f f 7 ~ - --

T + x 7

T + T 1 7

TX 7

T~ 7

]P( r ) 8

8 P f; 8

(*Ii 3 8920

-T 1 3

13918 T ( z )

(T( z))zcZr 1 3 0

*(z) 1 5

C_(S) - 1 5

% 1 5

I

W S ) 1 5

Wo 16 ,29

mn 1 6

- S(G,Zgn) 18

CI 25

9-CI 25

S t a b I'(i) 27

- k 27 , 29

1 27 , 29 kred - 27 , 30 red 27 , 30

ki 29, (10915)

f i 29

a 32