Cayley graph enumeration [Master thesis]

64 Pages·2000·2.741 MB·English

by  Marni Mishna

CAYLEY GRAPH ENUMERATION
Marni Mishna
BMath, University of Waterloo, 19 98.
THESISS UBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTERO F SCIENCE
IN THE DEPARTMENT
OF
MATHEMATIC&SS TATISTICS
@ Marni Mishna 2000
SIMON FRASER UNIVERSITY
March 2000
All rights reserved. This work rnay not be
reproduced in whole or in part, by photocopy
or other means, without permission of the author.
1+1 National Library Bibliothèque nationale
,cana& du Canada
Acquisitions and Acquisitions et
Bibtiogmphic Services semices bibliographiques
395 Wellington Street 395. nie Wellington
OrtawaON KtAON4 OüawaûN K 1 A W
canada Canada
The author has granted a non- L'auteur a accordé une licence non
exclusive licence allowing the exclusive permettant à la
National Library of Canada to Bibliothèque nationale du Canada de
reproduce, loan, distribute or sell reproduire, prêter, distribuer ou
copies of this thesis in microform, vendre des copies de cette thèse sous
paper or electronic formats. la forme de microfiche/film, de
reproduction sur papier ou sur format
électronique.
The author retains ownership of the L'auteur conserve la propriété du
copyright in this thesis. Neither the droit d'auteur qui protège cette thèse.
thesis nor substantial extracts from it Ni la thèse ni des extraits substantiels
may be printed or othenvise de celle-ci ne doivent être imprimés
reproduced without the author's ou autrement reproduits sans son
permission. autorisation.
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 so n 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 carenil1y 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 d elicious coffee automatically! does any mlxingj ob!
by solving cornplex muthematical fonnulas
Contents
Approval ii
Abstract iii
Acknowledgments
Contents
List of Tables
List of Figures
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 12
. . . . . . . . . . . . . . . . . . . . . .
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,an d 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 : CAYLEYG RAPH X(&, {r, r3, r2t))A ND CAYLEYD IGRAPH X(&, {r,r 2t))
graph X.
The next two definitions describe a Cayley graph.
DEFINITION. A Caylqsubset S ofa 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 as 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 pu pon
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 Dg.