COMBINATORIAL AND COMPUTATIONAL ASPECTS OF FINITE GEOMETRIES

Luis Armando Dissett

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy

Graduate Department of Computer Science University of Toronto

Copyright @ 2000 by Luis Armando Dissett

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Abstract

Combinatorial and computational aspects of finite geometries

Luis Armando Dissett

Doctor of Philosop hy

Graduate Department of Computer Science

University of Toronto

2000

This t hesis presents a methodology for the systematic generation of strongly regular

grap hs by fusing classes in large association schemes; t his met hodology includes:

an algorithm for finding the eigenvalue matrix of an association scheme;

an algorithm for determining the lattice of matrices determined by subgroups of

the automorphism group of an association scheme;

an integer programming formulation that allows us to determine quickly whether a

matrix (obtained by fusing the columns of the eigenvalue matrix of an association

scheme according to some group) will yield strongly regular graphs with a given

parameter set;

0 an exhaustive search algorithm that, given the parameters of a graph and a matrix,

will find al1 the combinations of columns of the matrix that correspond to strongly

regular graphs with the given parameters.

Using the methodology presented here, we have settled the existence problem for

strongly regular graphs with the following parameters:

lie have also found new graphs for many parameter sets for which some graphs were

already known, as well as graphs which were themselves known.

Many of the graphs found are pseudo-geometric, i.e., their parameters correspond to

those of the point graph of some hypothetical finite geometry. Some of the graphs are

actually geometric, and it is possible to recover the geometry from the maximum cliques

of the graph.

The graphs presented here are al1 of cyclotomic type; therefore they al1 can be ex-

pressed as two-intersection sets in suit able projective geometries, and t hey al1 determine

two-weight codes, but the methodology is not restricted to this kind of graphs. Several

results concerning two-intersection sets are derived.

We have found six interesting non-isomorphic strongly regular graphs with parameters

(625,156,29,42). If we consider the corresponding two-intersection sets in PG(3,5), the

affine lines determined by each of these sets (which are 5-cliques in the graph) form a

geometric structure that resembles a semipartial geometry: given any anti-0ag (p, L),

the number of points in L which are collinear with p is either O, 1 or 2 (Le., this number

takes one of three values, while in the case of partial geometries it takes only one value

and in the case of semipartial geometries it takes one of two values).

Another interesting property of these graphs is that it is possible to factorize the com-

plete graph Kszs into copies of some of these graphs; we exhibit three such factorizations,

each of them cyclic.

iii

We have also developed dgorithms to perform computations in a projective geom-

etry; the idea in which these algorithrns are based is to represent each d-dimensional

subspace of PG(n, q) by a canonical b u i s (set of d + 1 points that span the subspace);

this canonical basis can be obtained - from any other basis - by a process based in

Gaussian elimination.

To my wife and children.

Acknowledgements

First and foremost, my gratitude goes to my supervisor, Rudi Mathon, for showing

me the beauty of Combinatorics in generai, and of Finite Geometry in particular. His

support and advice during my studies were invaluable, and his contagious enthusiasm for

research \vil1 always be an example to imitate.

1 must also thank the members of my Thesis Commit tee: Derek Corneil, Eric Mendel-

sohn, F. Arthur Sherk and, in the last stage, Mike Molloy. They helped me al1 throughout

the process wit h t heir comments, suggestions and corrections.

My external examiner, Prof. hlex Rosa, deserves a speciai mention. His thorough-

ness in reading the final manuscript of the thesis found many rninor (and some major)

problems; this thesis would be substantially weaker without his conscientious revision.

Several persons in the Department of Computer Science have been very helpful in

many ways; the first names that come to my mind are Christina Christara, Jim Clarke.

Martha Hendriks, Teresa Miao, Lisa De Caro, Winnie Green, and Kathy Yen. Thanks to al1 of them for doing such a great job.

Life in a shared office can be great or miserable depending on the officemates one

has. 1 was lucky to share my office with three great persons: Robert Enenkel, Tomoyuki

Yamakami and Wayne Hayes. Wayne deserves a special mention for his patience, as he

has put up al1 these years with my messy nature (although in exchange, he has got first

hand esperimental verification of the Second Law of Thermodynamics).

Anyone who has written a thesis using T a or knows how much we owe to Don

Knuth and Leslie Lamport. Those of us in this department must also thank our local

T#/BTH guru: François Pitt, who wrote the U of T thesis style file, saved me and

many other graduates countless hours of struggle to comply with the style regulations

dictated by the School of Graduate Studies.

1 lost count of al1 the interesting people that 1 met in the Department in d l these years.

For s is years of my life 1 have been surrounded by one of the best social environments

1 could have asked for. 1 will certainly miss both the technical and non-technical con-

versations that 1 had with (at least): Dimitris Achlioptas, Geneviève Arboit, Frank van

Bussel, Alfredo Gabaldon, Theodoulos Garefaiakis, S herif G hali, Neil Graham, Wayne

Hayes, Carlos Hurtado, Mike Hutton, Eric Joanis, Valentin Kabanets, Iluju Kiringa, An-

tonina KoIokolova, Albert Lai, Keju Ma, Ali Mahmoodi, George Mihaila, David Mitchell,

David Modjeska, Lucia Moura, Hazem Nassef, David Neto, Brian Nixon, Alberto Pac-

canaro, Daniel Panario, Richard Pancer, Ioannis Papoutsakis, NataSa Peulj, François

Pitt, Faisal Qureshi, Yuri Rabinovich, Flavio Rizzolo, Sebastih Sardifia, Eric Schenk,

Alan Skelley, Mikhail Soutchanski, Eugenia Ternovskaya, Panayiotis Tsaparas, Gordon

Turpin, Yannis Velegrakis, Tomoyuki Yamakami and Daniel Zilio. Sorry, 1 must have

left out at least another forty people who should be here.

?VI any institutions supported me financially; without them it would have been impos-

sible to even think about completing (and in some cases, starting) my doctoral studies.

They are: the Faculty of Mathematics and the Vicerrectoria Académica (Academic Vice

Presidency) of my home university, the Pontificia Universidad Catdica de Chile, for their

continuous support throughout al1 these years; the Ministry of Planning (Mideplan) of

Chile; for the "Presidente de la Republica" (President of the Republic) scholarship that

they gave me between September 1993 and March 1997, including my tuition for three

years, my family's retum tickets and my ticket to corne to Toronto; the Organization

of American States, for their scholarship #53986 between September 1997 and August

1999, including my tuition for those two years and my ticket to go back to Chile; the

School of Graduate Studies at the University of Toronto, for a Bursary that they granted

me in 1996; and the Department of Computer Science, for a full Differential Fee Waiver

in 1996 and a partial one in 1999.

This acknowledgements' page would not be cornplete without a mention to those

who encouraged me to pursue graduate studies in Mathematics (for my Master's degree)

and Computer Science (for rny Ph.D.). 1 must thank the people from the Faculty of

Mathematics and the Department of Computer Science at the Pontificia Universidad

Catdica de Chile who always had confidence in my success (even at the times when

I did not). My gratitude goes to Renato Lewin, Claudio Fernhdez, Irene Mikenberg,

Vicky Marshall, Manuel Elgueta, Yadran Eterovic, Alvaro Campos, Ignacio Casas, and

especially to the memory of Rolando Chuaqui.

Finally, 1 must thank my family for their love, support, encouragement and patience.

Coming to Canada to fulfill my dream of getting a Ph.D. %-as the wildest adventure that

ive al1 ever imagined to engage in, and there they were, supporting me each step of the

way.

vii

Contents

O Introduction 1

1 Background 4

. . . . . . . . . . . . . . . . . . . . . . . 1.1 Prerequisites and basic concepts 4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Prerequisites 5

1.1.2 Some basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Projective geometries 7

. . . . . . . . . . . . . . . . . . . 1.2.1 Duality in projective geometries 8

. . . . . . . . . . . . . . . . . . . 1.2.2 Spreads in projective geometries 9

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Affine geometrîes 9

1.3.1 Nets and transversal designs related to affine planes . . . . . . . . 9

Examples of Projective and Affine Geometries . . . . . . . . . . . 9

. . . . . 1.4.1 Coordinatization of Desarguesian Projective Geometnes 12

. . . . . . . . . . . . . . . . . . . Collineations in a Projective Geometry 12

. . . . . . 1.5.1 The group of collineations of a Desarguesian geometry 13

. . . . . . . . . . . . . . . . . . . . . . . . . . . . Strongly regular graphs 14

1.6.1 Necessary conditions for the existence of strongly regular graphs . 14

1.6.2 Latin, pseudo Latin, and negative Latin square type graphs . . . . 17

. . . . . . . . . . . . . Partial geomet ries: definition and basic properties 20

. . . . . . . . . . . . . . . . 1.7.1 Basic properties of partial geometries 21

. . . . . . . . . . . . . . . . 1.7.2 The four classes of partial geometries 22

. . . . . . . . . . . . . . . . . . . . . . . . . . . . Semipartial geometries 22

. . . . . . . . . . . . . 1.8.1 Basic properties of semipartial geometries 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Association schemes 24

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.1 Definition 24

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.2 Basic properties 26

viii

. . . . . . . . . . . . . . . . . . . . . . . 1.9.3 The Bose-Mesner algebra 26

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.4 Group schemes 27

1 .9.5 Example: the cyclotomic schemes . . . . . . . . . . . . . . . . . . 28

2 Fusing Classes in Association Schemes 30

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The fusion problem 30

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Cyclotomic Graphs 32

2.2.1 The eigenvalues of the cyclotomic schemes . . . . . . . . . . . . . 32

. . . . . 2.2.2 Cyclotomic graphs as point sets in a projective geometry 35

. . . . . . . . . . . . . . . . . . 2.3.3 Duality and two-intersection sets 38

. . . . . . . . . . . . . . . . 2.2.4 Pairs of disjoint two-intersection sets 39

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Constructions 43

3 Computational Aspects 45

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Some common issues 45

. . . . . . . . . . . . . . . . . . . . . . . . 3.2 Other computational problems 46

. . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Graph isomorphism 46

3.2.2 Some ubiquitous NP-hard problems . . . . . . . . . . . . . . . . 47

. . . . . . . . . . . . . . . . 3.2.3 Computing in a projective geometry 47

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The main algorithms 52

3.3.1 .4 n algorithm to find the eigenvalue matrix of an association scheme 53

. . . . . . . . . . . . . . . . 3.3.2 An algorithm to find smailer matrices 54

3.3.3 An integer programming formulation for the fusion problem . . . 56

. . . . . . 3.3.4 An exhaustive search algorithm for the fusion problem 58

4 Computational results 61

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The lattice of matrices 62

. . . . . . . . . . 4.1.1 Example: the cyclotomic scheme on 625 vertices 62

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The graphs found 65

5 Analysis of some of the graphs found 68

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Cliques and subspaces 68

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Geometries 69

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Spreads in geometries 70

5.4 Factorizations of the complete graph into strongly regular graphs with

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . given parameters 71

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 The graph KZS6 71

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 - 4 2 Thegraph Ksw 72

6 Conclusions 76

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Contributions 76

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Future work 76

A Graphs on new parameter sets 78

B The new graphs 92

List of Tables

4.1 Lattice of matrices for C(625. 4) . . . . . . . . . . . . . . . . . . . . . . . 63

4.2 Number of matrices at each level of the lattice . . . . . . . . . . . . . . . 65

4.3 Information about the graphs found . . . . . . . . . . . . . . . . . . . . . 66

. . . . . . . . . . . . . . . . . A.1 Graphs with parameters srg(625.156.29.42) 78

. . . . . . . . . . . . . . . A.2 Graphs with parameters srg(729.308.127.132) 79

. . . . . . . . . . . . . . . .4.3 Graphs with parameters s~g(729~336~153~156) 82

. . . . . . . . . . . . . . . . X.4 Graphs with parameters srg(1024.231.38.56) 85

. . . . . . . . . . . . . . . . A 5 Graphs with parameters s~g(1024.297.76~90) 85

. . . . . . . . . . . . . . . .4.6 Graphs with parameters s~g(1024.330.98~110) 85

. . . . . . . . . . . . . . . .4.7 Graphs with parameters s~g(1024~363.122.132) 86

. . . . . . . . . . . . . . . A.8 Graphswithparameterss~g(l024.396~148~156) 87

. . . . . . . . . . . . . . . A.9 Graphs with parameters s~g(1024.429.176~182) 87

. . . . . . . . . . . . . . . -4.10 Graphs with parameters srg(1024.462.206.210) 88

List of Figures

1.1 The pair of triangles PQR. P'Q'R' is central. and also axial . . . . . . . . 11

. . . . . . . . . . . . . . . . 2.1 The adjacency in the graph determined by S 38

. . . . . . . . . . . . . . . . . . . . . 4.1 The Iattice of matrices for C(625,4) 64

xii

Chapter O

Introduction

The main goal of our thesis work has been to search for new finite geometries, specially

partial and semipartial geometries. Our interest in finite geometries is born mainly from

the intrinsic beauty of the topic, but also from the many relations between this area

and many other fields in mathematics and computer science: we find links between finite

geometries and group theory, graph t h e o l , coding theory and design theory. From a

more practical point of view, finite geometries are interesting because of their many

applications to such unrelated areas as cryptography (41, computer graphics [27, pp.

201-2081, and even computer architecture design [41].

X pervasive concept in the study of finite geometries (and combinatorics in general) is

that of strongly regular graphs: these appear as the point graphs of partial and semipartial

geometries, as the intersection graphs in certain designs, as distance graphs in two-weight

codes, to name only a few of the contexts in which they are found. For us, the main

importance of strongly regular graphs is that if such a graph is the point graph of a finite

geometry, then the lines of the geometry correspond to cliques in the graph; therefore, a

good first step in the search for a finite geometry is to search for strongly regular graphs

mhich are candidates to be the point graph of such a geometry.

This thesis presents a methodology for the systematic generation of strongly regular

graphs by fusing classes in large association schemes; this methodology is based mainly

on algorithms to determine suitable subsets of a large search space, where an exhaustive

search is guaranteed to find solutions, and to perforrn the exhaustive search itself once

these smaller search spaces have been deterrnined. This methodology has succeeded in

finding strongly regular graphs for parameter sets for which no graph was previously

known, as well as new graphs with parameters already known to be realizable. Many of

the graphs found are pseudo-geometric, i.e., their parameters correspond to those of the

point graph of some hypotheticd finite geometry. The graphs presented here can ail be

expressed as tw&ntersection sets in suitable projective geometries, but the methodology

is not restricted to this kind of graphs.

The structure of the thesis is as follows:

Chapter 1 gives a summary of the background materid needed to understand the

rest of the thesis. Most of the theorems are mentioned without proofs, and many ideas

are developed only at a superficial level; however, for the benefit of the reader, several

references are given.

Chapter 2 studies in detail the idea of fusing classes in association schemes to ob-

tain strongly regular graphs; as a concrete example, the graphs of cyclotomic type are

presented. We compute the eigenvalues of the cyclotomic association schemes, in a way

similar to Delsarte [2 11, and show the equivalence between cyclotomic graphs (Le., graphs

found by fusing classes in cyclotomic association schemes) and two-intersection sets in

projective geometries, and we show ways in which new tw+intersection sets can be de-

rived from old; in particular, we show that the dual of a two-intersection set is again a

two-intersection set, and that in certain cases ( e g , if the sets correspond to graphs of

pseudo Latin or negative Latin square types) the union of two such sets is again a two-

intersection set. The former has been shown in [16]; the latter appears to be a new result.

We conclude this chapter with some examples of the known constructions of graphs of

cyclotomic type.

Chapter 3 discusses some issues of algorithrnic nature that arise during a combinatorial

search process, and presents some new algorithms that are useful when searching for

strongly regular graphs. In particular, we show:

an algorithm for finding the eigenvalue matrix of an association scheme;

an algorithm for determining the lattice of matrices determined by subgroups of

the automorphism group of an association scheme;

an integer prograrnming formulation that allows us to determine quickly whether a

matrix (obtained by fusing the columns of the eigenvalue matrix of an association

scheme according to some group) will yield strongly regular graphs with a given

parameter set;

an exhaustive search algorithm that, given the parameters of a graph and a matrix,

will find al1 the combinations of columns of the matrix that correspond to strongly

regular graphs wit h the given parameters.

The three new algorithms mentioned above are Our own, and together with the integer

programming formulation (suggested by Lucia Moura), form the core of the methodology

developed in this work.

Chapter 4 summarizes the results from the application of Our algorithms t o some

cyclotomic schemes; we show - through an example, the cyclotomic scheme C(625,4)

- the structure of the lattice of matrices formed by fusing rows and columns in the

original matrix. We also give information on the parameters of some of the graphs

found, as well as the number of non-isornorphic graphs obtained in each case.

Chapter 5 is devoted to the analysis of some of the most interesting graphs found: we

show some (known) geometnc graphs that have corne up in our search, as well as six in-

cidence structures with geometric properties resembling those of semipartial geometries;

these are found by taking the linear representations determined by the six non-isomorphic

two-intersection sets in PG(3,5) that correspond to strongly regular graphs with param-

eters (625,156,29,42). We also show how copies of some of these graphs completely

partition the edge set of the complete graph on 625 vertices K s 2 ~ . Chapter 6 contains the conclusions that we have drawn from Our research, as well as

a summary of Our contributions and a list of the tasks that should be part of subsequent

research.

Appendix A lists some additional information about each of the graphs found with

parameters for which no graph was previously known, For each graph we list some

information, such as: size of the maximum clique, number of such cliques, size of the

automorphism group, number and sizes of the orbits determined by the stabilizer of a

point in the automorphism group.

Finally, appendix B shows, for each of the new parameter sets, some graphs with

those parameters (space considerations prevent us from giving a complete listing). We

present each graph as a tw~intersection set of points in a suitable projective geometry;

this appears to be the most compact representation possible.

Chapter 1

Background

In this chapter we give the definitions and basic properties of the objects that will be

needed in the remainder of this thesis. We do not give proofs of the various claims and

theorems mentioned in this section, but we give enough pointers to the literature for the

interested reader to find ail the sophisticated proofs (and many of the simpler ones). We

start by listing the prerequisites needed to understand the subsequent discussion, giving

references to textbooks where this materid can be found, and define some of the basic

concepts which are used without definition in the rest of this thesis. After this we define

projective and affine geometries, which provide a setting for most of Our discussion. We

continue with strongly regular graphs, important because of their relation to many kinds

of cornbinatorial objects: partial and semipartial geometries, combinatorial designs and

linear codes, among others.

After st rongly regular graphs, we focus on partial and semipartial geometries.

We conclude this chapter with a brief introduction to association schemes, which are

a generalization of the idea of strongly regular graphs.

1.1 Prerequisites and basic concepts

In this section we list the prerequisites and define some of the basic concepts that wiIl

need to be known in the subsequent discussion.

1.1.1 Prerequisites

We assume a basic knowledge of the following subjects; for each we give one or two

references:

0 Linear algebra, see [35];

r Group theory, in particular finite permutation groups; see [34];

r Finite field theory, see [42];

0 Graph t heory, see [7] and [XI; for the algebraic aspects of graph theory see also [5].

Some additional reference books that can be useful are (31, [23] and [52].

1.1.2 Some basic concepts

The foilowing concepts will be useful in the remainder of this thesis:

incidence s t ructure .4n incidence structure is a triple S = (p, 23, I ) where p, 23,I are

sets wïth

p n Z 3 = O a n d I ~ p x B .

The elernents of p are called points, those of 23 b1ock.s or lines, those of I flags, and

those of p x 23 - I anti-jlags. Instead of writing (p, B) E I we mi te pIB; instead

of writing "(p, B ) is a flag" we write "p and B are incident", "p is on B" or "B

passes through p" .

MTe wouid like to identifjr each block with the set of points incident to it. This would

simplify the notation, allowing us to use set-theoretic notation, such as " p E B"

instead of pIB, or (if S C p), "S B" instead of "every point of S is incident to

B", where B E B. We are able to do this without problems if no two blocks in

B have the same set of incident points. If this is not the case, we can still make

the identification, provided that we define 23 as a muftiset, i e . , a set with "rep-

etitions" allowed (otherwise, we will run into inconsistencies, e-g., when counting

"the number of blocks incident to a given point").

dual of an incidence structure Given an incidence structure S = (p, 23, 1), its dual

is the incidence structure S' = (23, p, 1-').

point g r aph of an incidence structure Given an incidence structure S = (p, B , I ) ,

consider the graph with vertex set V = p, and where two vertices x, y are adjacent

iff there exists a block B E 93 such that both x and y are incident to it.

The point graph of an incidence structure is also called the collinearity graph of

the structure.

resolvability A resolution of an incidence structure S = (p, 23, 1) is a partition of 23

into disjoint classes, where the blocks in each class (if identified with the sets of

points incident to them) partition p.

t-design A t-design with parameters (v, k, A) is an incidence structure S = (p, <B, I)

where Ip( = v, IBI = k for every B E 8, and where given any t-subset S of p, there

are exactly X blocks B E 23 such that S C B (here we are using the simplified

notation mentioned in the definition of incidence structure above).

b d a n c e d incomplete block design (BIBD) -4 balanced incomplete block design

BIBD(u, k, A) is a 2-design with parameters (v, k, A).

Steiner sys tem A Steiner system S(t, v, k) is a t-design with parameters (u, k, 1).

transversal design A transversal design TD(k , n) is an incidence structure S = (p, 23, I)

together with a partition of p into k n-sets1 in such a way that every block intersects

each set in the partition in exactly one point, and given two points in different sets

from the partition there is exactly one block containing both of them. From the

conditions it is easy to conclude that Ipl = nk, 1231 = n2.

net -4 net of order n and deficiency d is the dual of a TD(n + 1 - d, n). For the basic

theory on nets, see [13] and [14].

graph isomorphism Let G = (V, E ) and G' = (V', Et) be two graphs. An isomor-

phzsm between G and G' is a bijection f : V + V' such that for every v, w E V ,

{f (4, f (4) E E' - {v, w} E E-

graph automorphism -4n automorphism of a graph G is an isomorphism between the

graph and itself.

' These sets are cornmonly called "groupsn; a very bad choice of terminology, since the word here is not used in the normal mathematical sense but just in the everyday sense of "aggregationn. Maybe a better term would be "classesn.

linear code An [n, k] linear code over F, is a k-dimensional subspace of q. The ele-

ments of this subspace are called wdewofds.

weight of a codeword The weight of a codeword in a linear code is the number of

non-zero components.

two-weight code A two-wezght [n, k] w d e with weights wl and w* is an [n, k] linear

code where the weight of each non-zero codeword is either wl or wz.

1.2 Projective geometries

Consider an incidence structure S = (P, Ç, I ) satisfying the following axioms:

(PGl) For any two points p, q E P, there is a unique line L f C such that pILlq . This

line witl be denoted by p + q.

(PG2) Every line is incident with at ieast three points.

(PG3) If two distinct lines L, M are concurrent a t the point p, and q, r, s, t are four distinct

points in P - {p) such that qILIr and s I M I t then q + s and r + t are concurrent.

Xotice that from (PG1) and (PG2) we can conclude that no two lines have the same

set of incident points, and therefore (see the definition of incidence stmcture in 1.1.2) we

can identifi each line with the set of points incident to it.

A subspace of S is a subset S of P such that p, q E S p + q E S. We will consider projective geometries as defined in [23, Sec. 1.41, i.e., as the system

of a11 the subspaces of an incidence structure like the one described above.

From now on, when we refer to subspaces of G we mean the subspaces of the corre-

sponding incidence structure.

If S and T are subspaces of a projective geometry, then clearly S n T is a subspace.

More generally, if G is any family of subspaces of a projective geometry, then n6 is a

subspace.

We define the subspace spanned by a set of points in a projective geometry to be

the smalIest subspace containing that set. A set S of points is said to be independent

if no proper subset of S spans the same subspace as S. It can be proved that any two

independent subsets spanning a given subspace S must bave the same cardinality; we

cal1 this common cardinality the mnk r (S) of S. We define the dimension dim(S) of S

to be one l e s than the rank. Thus, subspaces of dimension O contain a unique point,

and subspaces of dimension 1 are the lines of the geometry2.

If the dimension of a projective geornetry is 2, it is called a projective plane.

If S and T are subspaces of a finite-dimensional projective geometry Ç, then

(this is called the rank formula). An immediate coroliary is that if L and L' are two

distinct lines contained in a common plane, then L n L' # 0. It is known that in a projective geometry, any two Lines have the same cardinality. If

this common cardinality is finite and equal to q + 1, we Say that the geometry has order

Q - If G is a finite projective plane of order q, then Ç consists of q2 + q + 1 points and

q2 + q + 1 iines, each point lies on q + 1 lines and each line contains q + 1 points. In other

words, a finite projective plane is a (symmetric) BIBD(q2 + q + 1, q + 1 , l ) . Conversely,

any symmetric BIBD(q2 + q + 1, q + 1 , l ) can easily be proved to satisfy the axioms for

a projective plane; therefore projective planes are precisely the symmetric BIBD's with

index 1.

1.2.1 Duality in projective geometries

It is easy to prove t bat the dual of a projective plane is a projective plane; more generally,

ive have the following theorem:

Theorem 1 (The principle of duality for projective planes). Gzven any theorem

ualid for al1 projective planes, its dual (i.e., the statement that results of interchanging

"points" and 'lines" in the original statement) is also valid.

In general, a projective geometry of any finite dimension d has a dual-although one

must be careful to interchange points and "hyperplanes" (maximal proper subspaces),

and, even more generaliy, r-dimensional and (d - r)-dimensional subspaces. Thus, we can

generalize the "principle of duality" to projective geometries of any (finite) dimension.

*Or, more precisely, the sets of points incident to each h e ; but as we said before, we identify both concepts.

1.2.2 Spreads in projective geometries

Definition 1. A t-spread in a d-dimensionai projective gwmetry Ç is a partition of the

set of points of Ç, where each set of the partition is a t-dimensional space.

-4 necessary condition for a t-spread to exist is that t + l must divide d+ 1. Conversely,

if t + 1 divides d + 1, a t-spread always exists, but only for small values of q, d and t have

ail the t-spreads of Ç been determined.

1.3 Affine geometries

An afine geometry is the set of al1 the subspaces of the form S - H where S ranges over

al1 the subspaces of a given projective geometry, and H is a fixed hyperplane of that

projective geometry.

If the projective geometry is called B, the affine geometry obtained by replacing each

subspace S of Ç by S - H (where H is a hyperplane) is denoted by GH. The concepts of rank and dimension of a subspace of an affine geometry are defined

in terms of the corresponding subspace of the projective geometry. The order of an &ne

geometry is the order of the corresponding projective geometry.

If the dimension of an affine geometry is 2, it wiil be called an a i n e plane.

Unlike the case of projective planes, the dual of an affine plane is not an affine plane;

in other words, the principle of duality is not valid for affine planes.

1.3.1 Nets and transversal designs related to affine planes

If we remove al1 the blocks from one parallel class of an affine plane of order q, we obtain

a TD(q , q) (the partition of the point set is given precisely by the removed blocks).

Conversely, if we have a TD(q, q) then, by adding blocks corresponding to the sets that

partition the point set of the design, we end up with an a n e plane of order q.

If instead we remove al1 the blocks from d parallel classes, we obtain a net of order q

and deficiency d.

1.4 Examples of Projective and Affine Geometries

The main source of examples of projective and affine geometries is the following con-

st ruction:

Let V be a d-dimensional vector space over the field K (d can be finite or infinite).

It is easy to verifi that the incidence structure S = (Pl L, 1) where

P = {W 5 V 1 dim(W) = 1)

L = {W 5 V 1 dim(M7) = 2)

I = {(W, W') E P x L 1 W 5 W ' )

satisfies the cuiiditious (PG1)-(PG3). We define P ( V ) as the projective geometry induced

by S. The r-dimensiona13 subspaces of S (for O 5 r 5 d - 1) are precisely the ( r + 1)- dimensiona14 subspaces of V. Equivalently, the projective-geometric concept of m n k

corresponds to the linear-algebraic concept of dimension.

Actually, the construction presented above can be generalized by considering "left

vector spaces" over 'bskew-fields" ; see [39] for the details.

The Desarguesian Property

In this and the following sections, we will assume that al1 the projective geometries have

dimension at l e s t 3, unless otherwise stated.

Let Ç be any projective geometry (not necessarily of the form P ( V ) ) . Three points P, Q and R of are said to form a triangle iff they are not collinear.

We will denote this triangle by PQR or APQR.

Let PQR and P'Q'R' be two triangles in Ç. We Say that they form a couple if the

points P, Pt, QI Q I , R and R' are six distinct points, and the lines L = P + Pl, L' = Q+Q1

and L" = R + R' are three distinct Iines.

-4 couple formed by the triangles P Q R and P'Q'R' is said to be central iff the lines

L! L' and L" are concurrent, and it is said to be azial iff the points P" = (Q+R)n(Q'+Rt),

Q" = (R + P ) n (R' + Pt) and R" = (P + Q) n (Pt + Q') are collinear.

Let us consider the condition

(Dl Every central couple of triangles is axial

It turns out that D is equivalent to its "dual" condition

(D') Every axial couple of triangles is central

See Figure 1.1.

In the pro jective-geornetrical sense. In the linear-algebraic sense.

Figure 1.1: The pair of triangles PQR, P'Q'R' iç central, and also axial.

A projective geometry G satisfjmg D (or equivalentiy D') is said to be Desarguesian

(since condition D is known as the Theorem of Desargues.) -4n affine geometry is said

to be Desarguesian iff the corresponding projective geometry is Desarguesian.

The importance of the Desarguesian property is demonstrated by the following two

t heorems:

Theorem 2. Al1 projective (and a f ine) geornetries of dimension > 2 are Desarguesian.

Theorem 3. If a projective geometry G is P(V) for some uector space V (over some

skew-field K) , then Ç às Desarguesian. Conversely, any Desarguesian projective geometry

is of the form P ( V ) for some suitable vector space V , over some skew-field K .

-4 corollary to the first of these theorems is that if Ç is a non-Desarguesian projective

geometry, then Ç m u t be a projective plane.

Remark 1. If dim(Ç) is finite, say n, we will denote G os P,(K) (recall that in this

case any n-dimensional vector space over K is isomorphic to Kn).

Remark 2. If K is finite, then K m u t be a field (th& is Wedderburn's Theorem, see

[25, p. 861 for a proof), i-e., K = F, where q = pn for some prime p and n E N. In this

case, if d is also finite, we d l denote Pd(Fq) as PG(d, q).

1.4.1 Coordinatization of Desarguesian Projective Geometries

Let Q = P ( V ) where V is a (d + 1)-dimensionai vector space over some skew-field K.

-Assume that a h e d basis B of V has been chosen.

In this case, each point P of Ç is a 1-dimensional subspace of V, and hence it can

be represented as ((xo,. . - , xd)) where xo, . . . , xd E K - {O} (here (xo,. . - , xd) are the

coordinates of some vector of P in the basis B). Without loss of generality, we can

assume that the last non-zero element in the ordered (d + 1)-tuple (xOi . . . , xd) is 1. In

the sarne way, the hyperplanes of Ç (which correspond to hyperplanes of V in the linear-

algebraic sense) can be represented as 1-dimensional subspaces of the dual space V' of

V' Le.' as ([ao, . - . , ad] ) where ao, . . . , ad E K - {O}. Again, without loss of generality,

we can assume t hat the 1 s t non-zero element of [ao, . . . , ad] is 1. -4 point ((xO , . . . , xd))

is in the hyperplane ([Q:. . . , ad]) iff aoxo + - - - + adxd = O, Le., iff the vector (xO,. . . , xd)

is in the kernel of the linear functional [ao,. . . ,ad]. If the reader prefers, he/she can

think of the vector [%, . . . , ad] as a vector "orthogonal to the hyperplane" in the (d + 1)-

dimensional space V: in this way, each vector (xo,. - . , xd) in the hyperplane naturally

satisfies aoxo + + a d x d = 0.

In the case where d = 2, the hyperplanes of Ç are the lines, and a point p = ((x, y, 2))

is in a line L = ([a, 6, cl) iff ax + by + cz = 0.

Remark 3. The coordinatization of a projectitle geometry = P ( V ) -where V zs finite-

dimensional- in such a zuay that the last non-zero coordinate of (xI, x2,. . . ,x,) is 1 is

called an homogeneous coordinatization of Ç. Homogeneous coordinates have applications

in Computer Graphies (see for example [27, pp. 204-2081).

1.5 Collineations in a Projective Geometry

One of the most important tools in -4lgebra and Combinatoncs is the study of the auto-

morphisms of the different structures we are interested in, or of the isornorphisms between

two such structures. In the case of projective geometries, we define automorphisms and

isomorphisms by requiring that inclusions between subspaces be preserved. It turns out

that a necessary and sufficient condition for a permutation of the point set of a projec-

tive geometry to be an autornorphism (or for a bijection between the point sets of two

geometries to be an isomorphism) is that it preserve collinearity, i.e., if P, Q are two

different points, then (P + Q)' = P' + Qd. Because of this, it is common to refer to

autornorphisms of (or isomorphisms between) projective geometries as "collzneations" . As is usually the case with many algebraic and combinatorial structures, the set of al1

the automorphisms of a projective geometry, together with the "composition" operation

is a group.

The groups of autornorphisms (collineations) of projective geometries, as well as t heir

most important subgroups (the so-called classical groups) have been extensively studied,

see for example [24] or [54].

1.5.1 The group of collineations of a Desarguesian geometry

Definition 2. Let V and V' be a left uector space ouer the isomorphic skew-fields K and

Kt respectively.

A function f : V + V' is said to be semi-linear iff the= as an isomorphism a : K -+ K' such that, giuen X E K and v, w E V , the following conditions hold:

2. f (Au) = X a f (v)

a 2s called the associated or accompanying isomorphism o f f . It is clear that i f K = Kt

and cu = 1 then f is a linear function from V to V'.

When V' = V , we say that f is a semi-linear transformation; in this case we will say

that a is the associated automorphism o f f . If f is a one to one and onto semz-linear

transfonnata'on then it is suid to be a non-singular serni-heur transfomation of V .

The importance of semi-linear transformations is given by the following theorem:

Theorem 4. If the geometry is Desarguesian, say P(V), then every collzneation is in-

duced by a sema-linear transformation of the vector space V .

The group of al1 the non-singular semi-linear transformations of V is denoted by

rL(V) . Two important subgroups of I'L(V) are the group GL(V) of al1 the linear

transformations of V, and, if K is a field, the group SL(V) of al1 the non-singular linear

transformations whose determinant5 is 1.

Each of these groups induces a group of collineations of Ç; they are denoted by

Pl? L ( V ) , PG L(V) and -if K is a field- PS L(V) respectively.

51n other words, the deteminant of its matrix in any basis, since this is an invariant if K is a field.

1.6 Strongly regular graphs

Our next object of study is the concept of strongly regular graphs, that we define now.

-4 good survey on strongly regular graphs is [l?], see also [ I l ] .

Definition 3. A strongly regular graph unath pammeters v , k , A and p (srg(v, k , A, p)

f o r short) is a graph G = (V, E ) where IV1 = u , the degree of each vertez is k (Le., G is

k-regular) and, giuen any 2, y E V ,

Claim 1. Gzven a srg(u, k , A, 1) G = (V, E ) , the complementary graph of G (i-e., the

graph G = (V,Ë) where Ë = {{x, y ) C V 1 x # y and { x , y ) # E ) ) zs also a strongly

regular graph. As for its pammeters, G is a

-4 proof of this clairn is simply a matter of counting.

Let G be a srg(v, k , X,p). If we denote by A the adjacency m a t r k of G, then the

conditions for G to be strongly regular can be written as follows:

1.6.1 Necessary conditions for the existence of strongly regular

graphs

We will give some necessary conditions for the existence of a srg(u, k , A, p) . We will

exclude disconnected graphs and their complements from our analysis.

Excluding these cases is e q u i d e n t to requiring O < p < k < v - 1, as the following

c h a h of claims shows.

Let G = (V, E ) be a srg(u, k, A, p). Then:

Claim 2. k ( k - A - 1 ) = (V - k - 1)p = kp.

Claim 3. p = O 28G às the disjoint union of wmplete graphs of order k + 1.

Claim 4. The condition "both G and G are connected" is equivalent to O < p < k < v- 1.

A first condition for the existence of a srg(u , k , A, p) cornes fiom the existence of the

complementary graph. Rom the parameters of this one we deduce that v - 2k + p - 2 1 O

is necessary.

A second condition arises from the study of the eigenvalues of the adjacency matrix

of G, A(G). It is known (see for example [5, pp. 14-15]) that if G is regular then k is

the Iargest eigenvalue of A(G). Moreover, if G is connected then the multiplicity of k is

1 ( the corresponding eigenspace being that generated by [l, . . . , llT).

Note 1. From now on, we will write A for A(G) .if the meaning is unambiguous.

The integrality conditions

Claim 5. Other than k , A h a two eigenvalues r and s ( r > s) given by

We cal1 f and g the multiplicities of these eigenvalues. Clearly, as the size of A is v

(and the rnultiplicity of k is l ) , f and g satisfy f + g = v - 1. Also, as the trace of A is -k(s+l) (k-S) O, we have k + f r + gs = 0; from these two equations we conclude that f = (k+rsl(r-s)

k(r+l)(k-r and g = (k+rs)(r-s)), and also that v can be expressed in terms of k , r and s as follows: (k-r)(k-s) u = k+rs -

Since f and g are multiplicities of eigenvalues, they must be integers, and hence they

impose the so-called integrality conditions on r and S. We can Say evea more about

integrality:

Claim 6. If f # g, then r and s m u t be integers.

The other case (Le., f = g) is called the half-case.

A partial converse to the integrality conditions is given by the following theorem:

Theorem 5. A regular connected gmph G is strongly regular iff its adjacency mat* has

three eigenvalues.

The Krein conditions

Consider a basis {j, XI, . . . , XI, yi, . . . , y,) of PI" consisting of eigenvectors of A, and where

j = (1, . . . , llT, the vectors xi are eigenvectors corresponding to the eigenvalue r and the

vectors y* are eigenvectors corresponding to the eigenvalue S.

Consider the vector space generated by the matrices A, I and J . This vector space is

closed under matrix multiplication; in other words, it is an algebnz (the so-called Bose-

Mesner algebra of the graph, see [9]), and another basis for this algebra (besides (1, A, J})

is {I,A,A2}. Let us denote this algebra by A.

Definition 4. The Schur product of two m x n matrices B and C is the m x n mat*

B 0 C defined b y

The matrices A. = 1, Al = A and A2 = J - I - A form a basis of A. As they are

clearly idempotent with respect to the Schur product, and the Schur product of any two

of them is the zero matrix, it follows that A is closed under the Schur product.

Let Eo, El and E2 be the matrices corresponding to the linear transformations defined

by

The matrices Ei are idempotents (Le., E: = El) , and they also satisfy EïEj = O

if i # j . They are called the minimal idempotents of A. Clearly, I = Eo + El + Es,

A = kEo + rEl + sE2, and A2 = k2Eo + r2El + s2E2. AS Il A and A2 can be expressed

in terms of {Eo , El , E2} , this is another basis for A.

As A is closed under the Schur product, there must exist numbers q:j such that

The numbers q:j are cailed the Krein parameters of the graph.

Let us denote by mi(O 5 i 5 2) the trace of Ei (i.e., no = 1, ml = f and m2 = g); 2

also, let Pi, be the numbers such that Aj = C PijEi for O 5 i 5 2, and let Sj be the i=O

sum of al1 the entries in the matrix Aj (i.e., So = v, SI = kv and S2 = V(V - k - 1)).

It can be proved (see [30, p. 227l)"hat

The Krein parameters must be non-negative (again, see7 [30, pp. 230-2311). These

constraints are automatically satisfied in almost al1 the cases; the only two cases where

t hey give non-trivial information are called the Kmin conditions (see [Il, p. 891):

These conditions can be rewritten as

Krein 1: (r + 1) (k + r + 2rs) 5 (k + r)(s + 1)2

Kreàn 2: (s + l ) ( k + s + 2 r s ) 5 (k +s)(r + 1)2

The absolute bound

-4nother relation that the parameters of a strongly regular graph must satisfy is the

so-called absolute bound (see [Il, p. 891):

1 1 v 5 ?f(f +3) and also u < - g ( g + 3 ) 2

1.6.2 Latin, pseudo Latin, and negative Latin square type graphs

Two interesting families of strongly regular graphs are those of the so-called, on the

one hand, Latin (and pseudo Latin) square type, and on the other hand, negative Latin

square type, which we define below.

Latin square and pseudo Latin square type graphs

Let u = n2, and suppose there are r MOLS (mutually orthogonal Latin squares) SI, . S..., Sr

of side n. Define a graph where the vertices are the pairs (2, j ) (where 1 5 i , j 5 n), and

6.4ctually, ~ i d s i l proves this for orassocialion sdiemcs, a generalization of the concept of strongly regular graphs that we will study in section 1.9.

7.4gain, this proof is more general, since it is for association schemes.

the adjacency relation is given by

( 2 , j) - (9: j t) (z = it) or ( j = jr) or 3k( l 5 k 5 r ) such that

entries ( 2 , j ) and (il, j') of Sc are equal;

ive cal1 this graph a Latin square type graph of type Lri2(n). Equivalently, if we take

g = r + 2, we can think of such a graph as the point graph of a net of order n and

deficiency (n + 1 - g ) , i.e., the dual of a transversal design with g parallel classes.

As an extreme, consider the case r = 0: the (unique) Latin square type graph of type

L2(n) is defined by the adjacency relation given by

this graph is called a square lattice graph. The other extreme case, namely r = n - 1,

corresponds to n + 1 MOLS of side n, or equivalently, to an affine plane of order n. It is

easy to see that in this case the graph (n) is the complete graph on n2 vertices.

C la im 7. A Latin square type graph of type L,(n) (with 2 5 g < n + 1) zs a

A sketch of the proof of this claim is as follows: consider a net with g parallel classes;

every point is in g lines and therefore is collinear to g(n - 1) other points; given two

collinear points 3: and y, there are n - 2 points in the line joining x and y, and in each

of the other g - 1 lines through x there are g - 2 points collinear to y (since out of the

g lines through y one joins it to z and another is parallel to the line being considered);

finally, given two non-collinear points x and y there are g lines through x and each of

these meets g - 1 of the lines through y (al1 of them except the one parallel t o it).

Any strongly regular graph whose parameters have the form given in (1.7) is called

a pseudo Latin square type graph; by abusing the notation we will say that i t is of type

LJn) for suitable g and n, even if the cormponding number of MOLS of side n does not

ezist. An example of the situation just descnbed is the following: although there are no

2 MOLS of side 6 [53], there exists a srg(36,20,10,12), which is of type L4(6).

Given a pseudo Latin square type graph of type Lg(n), its complement is also a pseudo

Latin square type graph, of type L,+i-g(n).

Negative Latin square type graphs

Mesner [48] proposed the following "major violation" of the assumptions made when

deducing the parameters of graphs of type L,(n) (narnely, n 2 1 and 2 5 g 5 n + 1 ) :

consider negative values for n and g, and use them in (1.7). If X = ( n - 2) + (g - 1 ) ( g - 2) 2 O then it is possible for these parameters to correspond to a strongly reguiar graph; Mesner

dubbed the class of such graphs "negative Latin square type graphs" . In order to avoid

using negative numbers, he defined a graph to be of type NL,(n) (with g, n E N) if its

parameters would correspond to a pseudo Latin square type graph of type L-,(-n). In

other words' a graph of t-vpe NL,(n) has parameters

Given a negative Latin square type graph of type NLg(n) , its complement is also

a negative Latin square type graph, of type NL,-I -g(n) . This imposes an additional

condition on g and n for the existence of the graph: not only X = (g + l ) ( g + 2) - n - 2

but also = ( n - g ) ( n + 1 - g ) - n - 2 must be nonnegative.

Negative Latin square type graphs are interesting to us mainly for two reasons:

1. Parameters for possible negative Latin square graphs are a high fraction of the

possible parameters for strongly regular graphs on a square number of vertices. To

quote Mesner [48, p. 5751:

. . . inspection of a list of arithmetically possible pararneters for two-

class association schemes leads to the interesting conjecture that when

u is a square, a high proportion of these pararneters fa11 in the group

divisible, L, and NL, series. As an illustration, in the range v 5 100, v

a square, there are at most 65 sets of two class parameters, of whicb 59

are in these three series.

If ive consider only graphs of type NL,, the situation is as follows: for up to

100 vertices, out of 32 possible parameter sets with a square number of vertices8,

14 correspond to graphs of negative Latin square type; for graphs of up to 1000

sThe discrepancy between our count of 32 versus Mesner's 65 is due to newer resuits on necessary conditions, that have decreased our estimate of the number of possible parameter sets, as weil as to the fact that we are counting complementary pairs of parameters only once; if we counted complements separately we would get 60 sets of parameters.

vertices, the proportion is 161 out of 418, and for up to 10000 vertices it is 1962 out

of 4939. Roughly, we can Say that for graphs of up to 10000 vertices the proportion

is about 40%.

2. For many parameter sets corresponding to graphs of negative Latin square type,

existence is an open question, and it seems to be somehow harder than for other

classes of parameters on a square number of vertices. In particular, this seems to

be the case when the number of vertices is a (square) prime power; for example,

for strongly regular graphs with v = 625, there are 7 parameter sets for which

existence is open, and 5 of these correspond to graphs of type NL,(25); for u = 729

7 parameter sets are open, and 3 of these are of type NL,(27); for v = 1024, 11

parameter sets are open, and 10 of these are of type NL,(32). In this thesis, we

show existence for severai of these families.

Another example that illustrates how difficult it is to decide the existence of graphs

of negative Latin square type is the following: in his original paper Mesner mentions

that, at the time of his writing, there were four cases with at most 100 vertices for

which existence was unknown, namely NL2 (6) , NL2 (7), NL3 (10) and N 4 (10).

Nowadays, thirty-two years later, only the first two cases have been settled, the

first in the affirmative and the second in the negative; the other two cases are still

open.

1.7 Partial geometries: definition and basic proper-

t ies

We proceed now to define and study partial geometries.

Definition 5. A partial geometry with aram met ers^ (s, t , a) (denoted by pg ( s , t , a)) is

an incidence structure S = ( P , C, I ) satisfying the following conditions:

1. Euery point of P is incident with (is in) t + 1 lines.

2. Every fine of L: is incident with (wntains) s + 1 points.

gHere we are foliowing the notation used by Thas [55]. Bose (81 and Brouwer and van Lint [Il] use ( K , R , T ) w h e r e K = s + l , R = t + l a n d T = a .

3. Given any line 1 and a point p & 1, the= are exactly a points in 2 which are collinear

with p.

1.7.1 Basic properties of partial geometries

If we cal1 v and b the number of points and lines in a partial geometry, by counting in

two ways, we find that v = ( s + q ( s t + a ~ and 6 = ( t+l st+a) Q

Thus, a necessary condition for the existence of a pg(s, t , a) is that <r must divide

both (s + 1)s t and (t + 1)st .

Clearly, the conditions defining a partial geometry are self-dual; t herefore, if we in-

terchange the points and the lines of a pg(s, t, a) we obtain a pg(t , s, a) which is called

the dual of the original partial geometry.

T h e relation between strongly regular graphs and partial geometries is given by the

following two daims:

C la im 8. Let S = ( P , C, 1) be a pg(s, t , a). Then the point graph of the geometry is

strongly regular, with parameters

Definition 6 (provisional). A strongly regular graph is said to be pseudo-geometric i f

its parameters are those of the point graph of a partial geometrylO.

Claim 9. If a strongly regular graph is pseudo-geometric with parameters corresponding

to a p g ( s , t , a) then, if C is a collection of edge-disjoint ( s + 1 ) -cliques that covers every

edge (i. e., C is a partition of the edges into ( s + 1)-cliques) then by wnsidering the cliques

of C as lines we get a pg(s, t , a), i. e., the graph is geometric.

From the parameters of the point graph of a pg(s, t, a) i t is possible to deduce that st s+l t + l the eigenvalues are s - a and -(t + 1) with multiplicities f = ,!s+ti(l-a; and v - 1 - f

respectively. From the fact that these multiplicities must be integers, we get

'OWe will later arnend this definition to include the point grapbs of semipartiai geometries, see Section 1.8.

Also, as the point graph must satisfy the Krein inequdities, we get

1.7.2 The four classes of partial geometries

Partial geometries can be divided into four (non disjoint) classes:

1. The partial geometries with a = s + 1 (or, dually, a = t + 1).

A partial geometry with a = s+l is the same as a BIBD(v , s+ 1,l) . In particular,

the pg(s, s , s + 1)'s are projective planes of order s, and the pg(s, s + 1, s + 1)'s are

affine planes of order S.

2. The partial geometries with a = s (or, dually, a = t).

A partial geometry with a = s is a transversal design TD(s + 1, t + 1); its dual

(i.e., a partial geometry with a = t) is a net of order t + 1 and deficiency s + 1 - a.

3. The partial geometries with a = 1.

.A partial geometry with a = 1 is called a generalized quadrangle. In this case, the

condition of Theorem 1.9 becornes (s + t + 2) 1 st(s + 1) (t + 1); this was also proved

by Feit and Higrnan in 1964 [26].

We denote a pg(s, t, 1) as GQ(s, t)

4. The partial geometries with 1 < a < min(s, t ) .

-4 partial geometry with 1 < cr < min(s, t) is called a proper partial geometry.

We will consider only generalized quadrangles and proper partial geometries. In the

nest subsections we present some of the known constructions of proper partial geometries

and generalized quadrangles. Most of this material is taken from [19].

1.8 Semipartial geomet ries

In this section, we briefly define and give some basic properties of semipartial geometries.

Semipartial geometries were introduced by Debroey and Thas [ZO]; the material presented

here can be found in [19, pp. 57-58] or directly in [20].

1.8.1 Basic propert ies of semipartial geometries

Definition 7. A semipartial geometry spg ( s , t , a, p ) with parameters s , t , a > O and p

is a n incidence structure S = (P, Cl 1) satàsfying the followàng conditions:

1. Every point o j P is incident wàth (is in ) t + 1 lznes.

2. Every line of L: às incident with ((contains) s + 1 points.

3. Given any lzne 1 and a point p 4 1, there are either O or a points in 1 which are

collinear with p.

4. Given any two non-collinear points, there are ezactly p poznts collinear with both.

Some consequences of this definition are the following:

1. The point graph of a semipartial geometry is a

2. Semipartial geometries are a generalization of, a t the same time, Cameron's par-

tial quadrangles [18] and partial geometries. A semipartial geometry is a partial

geometry if and only if p = ( t + 1)a. A semipartial geometry which is not a partial

geometry is called a proper semipartiai geometry.

3. The dual of a semipartial geometry is again a semipartial geometry only if s = t or

if it is a partial geometry.

As we mentioned in Section 1.7, we amend the definition of pseudo-geometric graph

given t here to the following:

Definition 8 (replacing Definition 6). A strongly regular gmph is said to be pseudo-

geometric if its pammeters are those of the point graph of a partial or semapartial georn-

etry.

The fact that the point graph of a semipartial geometry is strongly regular, plus some

counting arguments, imply the following theorem:

Theorem 6 (Debroey and Thas [20]). If S = (P, L, 1) is a proper spg(s, t, a , p)

then:

u t+ l 1. t 2 s, and therefom IPI = b = 2 v;

2. D = (t(a - 1) + s - 1 - p)2 + 4((t + 1)s - p) either quais 5 (in which m e S is

isomorphic to a pentagon) or is a square; in 60th cases,

is an integer;

3. a2 5 p < ( t + 1)o and a 1 p;

4 . p 1 ( t + l ) s t ( s + 1 - a);

5. a 1 ts(t + 1) and a 1 ts(s + 1);

6. a2 1 pst;

9. 3 1 v( t + l ) s (s - 1) and 3 1 v(t + l ) s t (o - 1);

1.9 Association schemes

1.9.1 Definition

Association schemes were introduced by Bose and Shimamoto [IO]. Several surveys and

introductory articles (e.g., 1221, [31], [33]) have been written on the subject; the material

on this section draws heavily on [12], but we only skim the surface of this topic; for

deeper results see the above cited works.

Bose and Shimamoto's original definition is as follows:

Definition 9. An association scheme with d classes is a finite set X together with d + 1 relations &, . . . , & on X , satisfiing:

( i ) {&, . . . , &) is a partition of X x X ;

(ii) & = {(x, x) : x E X) (the identity relation on X ) ;

(iii) each R, is symmetric, i-e., for ail X, y E X, if (x, y) E & then ( y , ~ ) E Ri-;

(iv) for each triple (il j, k) with O 5 2. j, k 5 d, there ezists a number P: such that for

any pair ( x , y) E R k , I{z : ( x , ~ ) E &, (GY) E Rj)I =&--

We define n to be 1x1. Conditions (i), (ii) and (iiz) can be interpreted as partitioning

the edges of the wmplete graph K, into the regular graphs ( X , &) (i > O). The ualency O of the graph ( X , f i ) is given by ni = p i , .

The numbers d, are calleà the intersection numbers of the scherne. W e define the

intersection matrices Lo, . , . , Ld to be given by

In particular, note that Lo = I (the identity mat*).

Delsarte [21] calls this a symmetric ussociation scherne; in his definition he replaced

(iii) by:

(iii7) for each i E {O, . . . , d} there exists a j E {O, - . . , d) such that whenever (x, y) E &,

( ~ ~ 5 ) E Rj;

(iii") for every i, j, k E {O,. . . , d ) , p: = p;i.

An even more general concept is that of a coherent configuration, introduced by

Higman [36, 37, 381. A wherent configuration requires (i), (iii'), (iv) and the following

weaker version of (iii):

(ii') {(x? x) : x E X) is the union of some of the &-'S.

If (ii) holds for a coherent configuration, then it is called homogeneous. We are

interested mainly in association schemes in the sense of Bose and Shimamoto; coherent

configurations wili be mentioned only tangentially.

An association scheme with two classes is equivalent to a pair of strongly regular

graphs; in this sense, association schemes are a generalization of strongly regular graphs.

1.9.2 Basic properties

Theorem 7 . The intersection nurnbers of an association scheme satisfy the following:

L m 4- Cl Pi jPk , = Cl P: ,PT- Conditions (i), (ii) and (iii) are straightforward. As for (iv), it c m be obtained by

counting in two ways the number of quadruples (w, x, y, r ) with (w, X) E R, (z, y) E Rj,

(y, z ) E Rt for a given fixed pair (w, z ) E &.

1.9.3 The Bose-Mesner algebra

Given an association scheme with d classes, define the n x n matrices Ai (1 < i 5 d) as

the adjacency matrices of the graphs (X, R), i.e.,

1 if (x, y) E &

O otherwise.

Rewriting (i)-(iv) in terms of these matrices, we get

3. Ai = AT for every i E {O, . . . , d } ;

4. for every i, j E {O,. . . , d l , AiAj = ~ f = ~ $ ~ ~ k .

Condition (i) implies that the matrices Ai are Iinearly independent; (ii)-(iv) imply that

t hey generate a (d + 1)-dimensional algebra of syrnrnetnc matrices wit h constant diagonal.

This algebra, that we will denote by A, is commutative, since Ai.4, = c : , ~ ~ ~ A ~ = d ~ , , , p ~ ~ ~ = AjAi. This algebra is called the Bose-Mesner algebra of the association

"The reader will notice that this algebra, as well as the following discussion, is a straightforward generalization of what we already saw in 1.6.1.

Since the matrices Ai commute, they can be diagonalized simultaneously (see [49]). It

follows ([El) that A has a unique basis of minimal idempotents Eo, . . . , En. The matrices

Ei satisfy

Clearly the matrix (l/n) J is a minimal idempotent; we will take Eo = ( l /n ) J. Let

P and (l/n)Q be the matrices relating the two bases {Ao, . . . , Ad) and {Eo, . . . , Ed) for

A, Le., let P and Q be defined by

Then, clearly, A,Ei = PijEï, which implies that the entries in the i-column of P

are the eigenvalues of Aj and the columns of Ei are the corresponding eigenvectors; in

virtue of this, we Say that P is the eigenmatriz of the scheme. -&O, it is easy to deduce

that PQ = QP = nI; this suggests that a certain kind of duality relation is a t play

between the matrices P and Q. In particular, this raises the question of whether there

esists an association scheme for which the roles of the matrices P and Q are reversed,

i.e., an association scbeme having Q as eigenmatrix. Two association schemes having

eigenmatrices P and Q with P Q = QP = nX are said to be fonnal duals of each other.

In general, little is known about this (formal) duality.

We will not go on to study in greater detail the relations between P and Q; in par-

ticular, Ive wiil not study the general version of the Krein conditions that was mentioned

in 1.6.1, since we will not need it in the ensuing discussion.

1.9.4 Group schemes

A rich source of association schemes (and, more generally, of coherent configurations) is

given by certain permutation groups. More precisely, we have the following

Theorem 8. Let Ç be a permutation group acting on the finite set X . The orbits of

.X x X under are called orbitals. In geneml, the orbitais f o m a coherent configuration,

which is homogeneous t#G is transitive on X ; if in addition Ç acts generously transitive

on X (ie., if for every x, y E X there exists some element of Ç which interchanges x

and y) , then the orbitals fonn an association scheme.

Note that if a group acts generously transitive on a set X then the order of must

be even: if g E Ç interchanges x and y then for any odd exponent r, gr(x) = y and

therefore gr # 1; this implies that the order of g is even and this in turn implies that the

order of Ç is even.

Our first and on1yl2 example of association scheme, subject of the next subsection,

can be presented in this way.

1.9.5 Example: the cyclotomic schemes

We ilhstrate the concepts discussed earlier in this section by considering the so-called

cyclo tornic schemes.

In the rest of this subsection, let p be a prime, q = pm (m 2 1) and let e be a divisor

of q - 1. Consider the finite field K = IF,, and let a be a primitive element of K. We

define a coherent configuration on K as follows: for each j (O < j 5 e) the relation

Rj is giwn by (x, y) E R, iff x - y = a"+j form some 2. It is easy to prove that this

is an homogeneous coherent configuration; the simplest way to do this is to notice that

the relations Rj are the orbitals determined by the group A . C, where A is the additive

group of K (which is elementary Abelian of order q) and C is the cyclic group (of order

e) generated by the mapping z ct ~ a ( 9 - l ) / ~ . We denote this coherent configuration by

C(q, e) (the second parameter is the number of i-associates of a given element, for any 2).

The fact that this coherent configuration is homogeneous is clear since A (and therefore

also A C) is transitive on K.

Under what conditions is A C generously transitive on K? This is cleariy the case

if q is even (since for q even, A acts generously transitive on X: if there is a z E K such

that x + z = y, then y + r = (x + z ) + z = x + ( z + z) = x). If q is odd, then the order

of C must be even (since the order of A - C is even); it follows that C must contain an

element of order 2, and therefore e must be even. Conversely, if e is even, C contains

the permutation x ct xa(q-l)/* = -x, and therefore A C contains al1 the permutations

of the form x c) y - x with y E K. Now it is easy, given a, 6 E K, to find an element of

'* Besides strongly reguiar graphs.

-1 - C that interchanges a and 6; this element is x H (a + b) - x (which is obtained by

takhg y = a + b above).

Thus a necessary and sufficient condition for C(q, e) to be generously transitive (and

t herefore for A . C to induce an association scheme on K ) is that either q or e be even.

If q = pm is odd, we d l only consider cyclotomic schemes C(q, e) where ( p - 1) [ e; one

reason for this is that these schemes can be nicely represented as point sets in suitable

projective geometries.

C hapter 2

Fusing Classes in Association

Schemes

2.1 The fusion problem

Consider a d-class association scheme, represented by the adjacency matrices A, corre-

sponding to the relations Ri-

Recall that the Ajls commute and, therefore, they can be diagonalized simultane-

ously; also recall that the algebra A spanned by the A,'s has a unique basis of minimal

idempotents Eo, EL, . . . , Ed, and that the relation between the ,4i7s and the Ei7s can be

surnrnarized by A, = Cf=, PijEi.

-4s AjEi = PijE,, the Pij's are the eigenvalues of Aj and the columns of Ei are the

corresponding eigenvectors (we identify Ei wit h the subspace spanned by t hese vectors) . Let iI be a partition of (1, . . . , d), and for each set ?r E II replace the matrices

a l j : j E T by B, = C - 3- A,.

We are effectively "fusing" the classes corresponding to the elements of each 7r E II. The new matrices B, are stilt connected to the minimal idempotents of A by equations

similar to the ones before: d / \

In terms of rows and columns of the matrix P, we are adding together the columns of P

that belong to the same set in the partition II. Let Pt be the matrix that results from adding the columns of P according to ri, in

other words, PI:, = C . Pi,. 3-

The matrices B, will defme a IlIl-class association scheme on X iff the d (non-trivial)

minimal idempotents of A can be replaceci by a new set of minimal idempotents of car-

dinality Ill 1; these new idempotents are direct sums of some of the original idempotents,

and correspond to identical rows in the matrix Pt. The eigenvaiues of the new scheme

will be the numbers PA, correspondhg to the matrix Br and the idempotent associated

with the rows of P' identical to the i-th one.

Thus, in the general case, deciding whether a partition rI of { 1, . . . , d} yields an

association scheme with fewer classes is the same as deciding whether il partitions the

set of rows in the matrix P into IlIl sets of identical rows. In the particular case where

we want to obtain a strongly regular graph with eigenvalues r and s, III1 = 2, and it

suffices to decide whether xjEn Pij E {r, s} .

Of course, more interesting than to decide, given a partition n of the set (1, . . . , d),

whether it yields a strongly regular graph (or fewer-class association scheme) with given

parameters, is to find all possible partitions that do so.

This is no easy feat, for this problem (which we call the fvsion problem) is related

to the NP-complete1 problem known as SUBSET SUM: given a finite set A, a positive

integer B (the goal), and a size s ( a ) E Z+ for each a E A, does there exist a subset

-4' C A such that the sum of the sizes of the elements in A' is exactly B?

This relation is as follows: clearly, the fusion problem is an example of enurneration

problem. We call the related decision problern (i.e., given an association scheme and

the parameters for a possible strongly regular graph on the sarne number of vertices, to

decide whether there exists at least one partition Il satiskng the required constraints)

FUSION-E? -4s a decision problem, this is not harder than Our original enurneration

version of fusion. A reasonable generalization of FUSION-E can be stated as

Given two integers rn and n, an rn x n matrix P? and a set of 2m "goals"

{ g i k : 1 5 i $ m, k E {l, 2}), does there exist a partition iI of the set {l, . . . , n) into 2 subsets nl and nz = S-nl such that for every i, k such that 1 5 i 5 rn,

1 I k E { 1 7 217 xjExk Rj = gik?

It is easy to see that the SUBSET-SUM problem is a particular case of this: given a

set .4 with integer sizes s(a) for a E A, and a goal B, there is a subset A' of A the

sizes of whose elements add up to B if and onIy if the following instance of the above

'For an introduction to the theory of NP-completeness, we refer the reader to [29]. *Here, the "En stands for k t e n c e

problem has "YES" for an answer: take m to be 1, n = IAl, the only row of the matrix

P to be filled with the sizes of the elements of the set, and the two goals to be gll = BI

912 = (CiEA 2 ) - B- Of course, it might be possible that, because of the restrictions imposed on the

FUSION-E problem (namely, the facts that the number of rows and columns are not

independent, the "goals" for every column are the same, and the number of columns

adding up to a given goal must be the multiplicity of that eigenvalue of the graph), it

be solvable in polynomial time. It is also possible that the FUSION-E problem itself

(as opposed to the generalization presented above) be NP-complete, but a proof or a

refutation of this is unknown to us at present.

2.2 Cyclotomic Graphs

-4 strongly regular graph obtained by fusing classes in a cyclotomic association scheme

is caIled a cyclotomic graph. In this thesis we use the methodology presented above to

search for cyclotomic graphs of several orders. This does not mean that the methodology

is restricted to search only in cyclotomic schemes; we have decided to focus on this type

of schemes simply because they seemed more promising.

In the first part of this section, we study the eigenvalues of the cyclotomic schemes and

determine the groups of automorphisms of their eigenmatrices. After that? we present a

way to associate a cyclotomic graph obtained from C(qn,q - 1) with a set of points in

PG(n - I , q ) .

2.2.1 The eigenvalues of the cyclotomic schemes

Let p be a prime, q = pm (m 2 l), and let n > 1. We want to describe the eigenvalue

matrix of the cyclotomic association scheme C(qn, k(q - 1)) for k E N such that k(q - 1) 1 qn - 1.

It turns out that for any such k, the eigenvalue matrix can be obtained by 'fusing'

groups of k rows (and columns) of the eigenvalue matrix of C(qn, q - 1). This is why we

will concentrate Our efforts in this latter scheme.

In what follows, let H = IF9, and let f be an irreducible polynomial of degree n in

H [XI. We can describe K = IF9,, the finite field on qn elements, as H [x] / (f). Let a be

a primitive element in K.

Consider the vector space V QV" where the coordinates are indexed3 by elements of

K. A basis of V is given by the vectors {e, : a E K), where e. has a 1 in the coordinate

corresponding to a, and a O elsewhere.

The coherent configuration4 C(qn, 1) is defined by the qn matrices Ab : b E K, where

Ab(ea) = e.+b. Thus, A. is the identity, and A l , if we sort the elements of K according

to the additive structure of K, is a block-diagonal matrix with pmn-' diagonal blocks,

each a p x p matrix with the form

Since the characteristic polynomial of each of these blocks is xp - 1, the characteristic

polynomial of Al is (xp - l)pmn-' .

Every other matrix Ab (where b E K*) is similar to Al, the corresponding permutation

matrix is given by the permutation x H bx of K; therefore, each of these matrices has

the same characteristic polynomial as Al.

In order to get the scheme C(qn , q - 1) we must identify ( "fuse" ) the classes determined

by Aaj and A,t if j G k (mod (qn - l) /(q - 1)). As a ( * n - L ) I ( q - l ) is a primitive element

of H, the net effect of this is that if a, b are elements of Km, A, and Ab are considered

equivalent iff a /b E H*. In other words, we can define this scheme as follows:

For each 6 E K*, let Bb = ChEH. Abh.

Clearly, Bb = Bbh for any h E H*. In order to consider only one representative

from among al1 these matrices, i t suffices to consider the matrices B,j where O 5 j <

(qn - l ) / ( q - 1). We introduce the notations K/H and K * / H to denote respectively the sets {O} U

{d : O 5 j < (qn - 1) / (q - l)} and {d : O 5 j < (qn - l ) / ( q - 1)). This way, the above

set of matrices can be described as {Bb : b E K 1 / H ) .

3This indexing can be done in any order, but we will consider two of these to be special: on the one hand, we wiii l i t the elements of K in the order (O, 1,a , a2, a3,. . . , aqn-*); on the other hand, we will regard the elements of K as polynomials in H [z] / (f) and list the elements of K in lexicographie order of their coefficients. In a manner of speaking, the former ordering emphasizes the multiplicative structure of K, whiie the latter mirrors its additive structure. We will move between these (or even other) orderings, according to convenience.

4See Subsection 1.9.1.

It is easy to check that the matrices {Bb : b E K o / H ) , together with the identity

matrix Bo = A. = 1, do define an association scheme; in particular, each matrix Bb with b b E K * / H defines an equivalence relation - given by xAy iff (z - y)/b E H*; the matrix

O O Bo defines the equivalence relation - given by x-y iff x = y.

VVe now set out to find the eigenvalue matrix of this scheme; we follow the ideas of

Lemma 2 in 1561.

First, we identi& K with its additive group A ( K ) in the natural way; and by abusing

t h e notation, we denote the elements of A(K) by the corresponding elements of K.

The canonical character x1 of A(K) is given by x1(x) = e2"'.*(R(f)l* (here i is the

imaginary unit); any other character of A(K) can be represented as x., where a E K

and X. (2) = xi (ax) (for a brief explanation of character t heory see [42, pp. 187-1921).

Theorem 9. If we consider the characters X, of A(K) as vectors in 0" with indices

ranging ouer K , then these vectors are ezgenvectors of the matrices Ba : b E K/H. If b = O then the eigenvalue wrresponding to X, is 1; if b # O then the eigenvalue is q - 1

if Tdabh) = O for every h E H , and -1 othenuise.

Proof. Let a, x E K y b E K * / H (the case b = O is trivial, since Bb is the identity).

Let y = ab. When considered as a vector in QPn with indices ranging over K, the

If Tr(yh) = O for every h E H, this sum reduces to Ch,, 1 = q. Otherwise, consider

the linear functional L : H + Fp given by L(h) = 'It(yh). -4s this functional is not

identically 0, its kernel has dimension m - 1 and therefore there are pm-' elements

h E H for which Tr(yh) = O. Clearly, for each f E F,, the set {h E H : L(h) = f) has

cardinality (since it is a coset of ker(L)), and therefore in this case ChqH xl(yh) = pm-' fEFp e2niflp which is O since the last sum is the surn of al1 the p t h roots of unity.

Finally, since Ch,,. xl(yh) = CIEH ~ l ( y h ) - 1, the proof is complete. O

Remark 4. As the vectors xa are mutually orthogonal, they are lzneurly independent

over @. As the number of chamcters is IKI = qn, these vectors form a b a i s of O".

Remark 5. If we leove out the O-th m w and the O-th wlumn ( w m p o n d i n g to the trivial

character ~0 and the identity matriz, respeetiuely), the entries of the eigenmatriz are in

a 1-1 correspondence with the point-hyperplane incidence matriz of PG(n - 1, q) (the

correspondence being given by: point a is incident to hyperplane b if the entry (a , b ) of

the mat* is q - 1).

Proof. The part of the matrix we are refemng to is given by

q - 1 if Tr(abh) = O Vh E H =

- 1 ot herwise,

where a, b E K*/H. If we fix b E K0/H, we need to prove that the set

is an (n - 1)-dimensionai subspace of K when considered as a vector space over H.

Clearly, Sb is a subspace of K; in order to prove that its dimension is n - 1 it suffices

to prove that lSbl is qn-'. We evaluate CZEK ChEH x1(xbh) in two ways: on the one

hand, it is xheH CzEK xl(xbh) = ChEH CzEK xZ(bh) = ChpH 0 = 0, and on the other

hand (using the claim above), it is (q - 1) ]Sb[ + (- 1) (qn - (Sbl), which implies jSbl = qn-l

as desired. 0

Corollary 1. Sirice the eigenmatriz P has the stmcture of the incidence matriz of PG(n-

1 , q ) , its autornorphisrn gmup is the projective group PFL, (q) , with order mqn(n-1)/211r-2(qi- - 1) (recall that m is such that q = pm).

2.2.2 Cyclotomic graphs as point sets in a projective geometry

In what follows, we present a simplified representation of the cycIotomic schemes in terms

of projective geometries. First, we define a concept that will be useful later:

Definition 10. A two-intersection set in PG(n - 1,q) as a proper, non-empty subset

S of PG(n - 1, q) such that there ezist two integers A and B, and euery hyperplane of

PG(n - 1, q) intersects S in either A or B points.

It is clear that the complement of a two-intersection set (relative to PG(n - 1, q)) is

also a two-intersection set.

Consider now the cyclotomic scheme C(qn,q - 1). We can identify the elements of

I F with the affine part of PG(n, q) . The set of z-th associates of z E Fqn is given by

{ z + na' : n E }, which (together with the point r ) is a line in AG(n, q). This line

determines a unique point in the hyperplane a t infinity in PG(n,q), which is a copy of

PG(n - 1, q); the homogeneous coordinates of this point (as an element of PG(n - 1, q)

are obtained by taking the coordinates of ai (as an element of q) and dividing every

coordinate by the last non-zero coordinate; this will make the 1 s t non-zero coordinate 1.

We 'ewill denote this point by [ai]. Notice that this point of PG(n - 1, q) is independent

of the choice of z (parallel lines in AG(n, q) intersect PG(n - 1,q) in the same point),

and thus we can Say that the point [ai] represents the i-th class of the association scheme

C W ? q - 1).

Thus, finding a cyclotornic graph by fusing classes in C(qn, q - 1) can be regarded

as partitioning PG(n - 1, q) into two sets of points (corresponding to the graph and its

complement, respectively). Cal1 S one of these sets; the adjacency in the graph is given

by the condition x - y iff the line joining x and y intersects S. It turns out that both S

and its complement are two-intersection sets, as the following claim proves.

Claim 10. If a subset S of PG(n-1, q) determines a cyclotomic graph with the adjacency

described above, then S is a two-intersection set. The size IV of S , as well crs the possible

cardinalities A and B of the intersections, are related to the eigenvalues of the gmph as

follows: if the eigenvalues of the graph are k (the valency), r and s , then we mvst have:

k = N(q - 1),

= A(q- 1) + (N -A)(-l) ,

s = B ( Q - l ) + ( N - B ) ( - 1 ) .

Proof. The first condition cornes from the fact that each point x in AG(n, q) is adjacent

to N ( q - 1) other points (q - 1 in each of the N Iines determined by x and the points of

the set S); the other conditions a i s e because the eigenvalues of the resulting association

scheme corresponding to a given hyperplane of PG(n - 1, q) will be the sum of (q - 1)

for each point of S which is in the hyperplane and -1 for each point of S not in the

hyperplane. Thus, if the size of the intersection between S and a hyperplane is C, then

the corresponding eigenvalue of the graph must be C(q - 1) + (N - C)(-1). As the

graph has only two eigenvalues (besides k), we must have exactly two conditions of this

type, Le., C can take exactly two values, which we cal1 A and B, corresponding to the

eigenvalues r and S. O

Solving the above system of equations, we get

How many hyperplanes intersect S in A points and hou7 many in B points? Let us

Say that r n ~ hyperplanes intersect S in A points and r n ~ intersect i t in B points. As

hyperplanes intersecting S in A or B points correspond to eigenvalues of the cyclotomic

scheme, these numbers are retated to the multiplicities f and g of the eigenvalues r and

s; more precisely, r n ~ = f / ( q - 1) and rns = g/(q - 1). We can also express m a and

r n ~ in terms of N, A, B, q and n: let us denote the total number of hyperplanes in

PG(n - 1, q) by 6, = (qn - l)/(q - 1) (and similarly, we will call bl = (ql - l)/(q - 1) for

O <_ 1 <_ n). We have

ma + m ~ = b,,

and as each point is incident to b,-' hyperplanes, by counting in two ways the pairs (p, h)

where p E S and h is a hyperplane of PG(n - 1, q) containing p, we get

Solving these two equations for r n ~ and r n ~ we get

B - N - qn-'(Bq - N) r n ~ =

( A - B)(q - 1) 7

A - N - qn-l(N - Aq) r n ~ =

( A - B)(q - 1)

Each of the sets defined above is called a two-intersection set in PG(n - 1, q) (see

figure 2.1), or, in the notation of [16], a projective ( N , n , A , B) set. The correspondence

between cyclotomic graphs and two-intersection sets was already shown by Delsarte [21]

and by Calderbank and Cantor [16, Thm. 3.21. Theorern 3.1 in [16] proves the equivalence

of two-intersection sets and projective two-weight codes by showing that to each two-

intersection set with parameters (N, n, A, B ) corresponds a projective two-weight [N, n]

code with weights N - A and N - B, and vice versa.

Figure 2.1 illustrates how the adjacency in the graph defined by S is determineci:

from this figure, we conclude that a is adjacent to both d and e, b is adjacent to e, and

c is adjacent to neither d nor e.

Figure 2.1: The adjacency in the graph deterrnined by S.

2.2.3 Duality and two-intersection sets

In Section 5 of [16], it is show that every two-intersection set in PG(n - 1, q) determines

a two-intersection set in the dual of PG(n - 1, q) . More precisely, we have the following:

Theorem 10. Let S be a two-intersection set &th pammeters ( N , n, A, B ) in PG(n - 1, q ) , and let S' be the set of hyperplanes of PG(n - 1, q) that intersect S in A points.

Let H denote the number of hyperplanes in PG(n - 1, q) containing a jixed line (or,

equiualently, a jked pair of points). Then, when considered as a point set in the dual of

PG(n - 1, q) , S' is a two-intersection set urith parameters (ma, n, A', B') = ((B - N - 4 " - ' ( B q - N ) ) / ( ( A - B ) ( q - l ) ) , n , ( H ( N - 1) -bn- i (B- l ) ) / (A-B) , (H(bn-N-1 ) - b,-l(b,-l - B - l ) ) / (A - B)) (recul1 that br = (qL - l)/(q - 1)).

Proof. Let p E S, and let A' be the number of hyperplanes of S that contain p. CIearly,

as there are bn-l hyperplanes through p, bn-l - A' hyperplanes of these intersect S in

B points. Counting in two ways the pairs of points of the form (p, r), with r CE S - p,

incident with hyperplanes of PG(n - 1, q), we find that

which implies that A' does not depend on p and is always (H(N-1) -6,-1 (B-l))/(A-B).

Analogously, given a point p E PG(n - 1, q) - S, p is incident with B' = (H (b, - N - 1) - bn-l(bn-l - B - 1))/(A - B) hyperplanes of S' (since PG(n - 1, q) - S is a

two-intersection set with parameters (b, - N, n, b,-l - A, bn-l - B)). As the points of

PG(n - 1, q ) are the hyperplanes in the dual space, S' is a twwintersection set with the

required parameters. O

2.2.4 Pairs of disjoint two-intersection sets

Let SI, S2 be two disjoint two-intersection sets in PG(n - 1, q). In general, their union

wilI not be a two-intersection set, since if the possible sizes of the intersections of Si with

hyperplanes are Ai and Bi, then S1 U S2 can intersect the hyperplanes in any of the four

sizes {Ai + A2, Ai + B2, Bi + A*, Bi f B2}. However, in the special cases when both Si and S2 correspond to graphs of pseudo Latin (or negative Latin) square type, their union

is again a two-intersection set corresponding to a graph of pseudo Latin (respectively,

negative Latin) square type. More precisely, if rn = qn, we have the following two

t heorems regarding two-intersection sets in PG(2n - 1, q):

Theorem 11. If SI , S2 are disjoint two-intersection sets in PG(2n - 1, q) corresponding

to two graphs of pseudo Latin square type L,(m) and Ly (m) respectively (where m = qn),

then Si u S2 is a two-intersection set corresponding to a graph of pseudo Latin square

type L,+# (4 -

Proof. CVe know that the parameters for a graph of pseudo Latin square type L,(m) are

v = m2; k = g(m - l ) , X = rn - 2 + (g - l)(g - 2), p = g(g - 1). The eigenvalues of this

graph are r = m - g and s = - g , with multiplicities g(m - 1) = k and (m - l)(m - g + 1)

respectively. From this we conclude that the size and two possible intersection numbers

for the corresponding tweintersection set are N = g(m - l) /(q - l), A = (q(m - g) + m ( g - l))/(q2 - q) and i3 = g(m - q)/(q2 - q). We will cal1 these numbers N,, A, and

B, in order to emphasize their dependence on g.

Simple algebraic computations show that:

It only remains to prove that no hyperplane in PG(2n - 1, q) intersects SI U S2 in

A, t il,, points.

Indeed: let x, y, z, w be the numbers of hyperplanes intersecting (SI, S2) in (A,, A#),

(A,, Bd ) , (B,, A#), (B,, B#) points respectively. We easily find the following three con-

dit ions:

x + Y = 9(m - - 11,

x + z = gt(m - l)/(q - l ) ,

z + y + r + w = (m2- l ) / (q-1) ;

the first two conditions are given by the multiplicities of the largest eigent-alues of the

graph, and the third one cornes from the total number of hyperplanes in PG(2n - 1, q).

In order to uniquely determine x, y, z, w, we need a fourth condition. This is found by

counting the triples (P l , Pz, n) (where Pi E Si and rr is a hyperplane containing Pl and P2) in two ways: on the one hand, if (as before) we denote by H the number of hyperplanes

in PG(2n - 1, q) containing a fixed line, then there are N, 1Vf - H such triples; on the

ot her hand, the total number of such triples is xA,Ad + yA& + rBgAd + wB,B#. As C -

it is known that El equals the Caussian coefficient L Z n l 2] = ( q 2 ~ - 2 - l>/k - u t We

9 obtain as a fourth condition

Solving the resulting system of equations, we get x = O, i.e., no hyperplane intersects SI in

-41 points and S2 in A2 points, as required. Incidentally, the values of the other unknowns

are y = g(m- l)/(q- l ) , z = g'(m-l)/(q-1) and w = (m- l)(m-g-gt+l)/(q- 1). 13

Theorem 12. If S1, S2 are disjoint two-intersection sets in PG(2n - 1, q) cowesponding

to two graphs of negative Latin square type NL,(m) and NL,(m) respectively (where

rn = qn), then Si U S2 iS a two-intersection set componding to a graph of negative Latin

square type NL, y (rn).

Proof. The proof is, mutatis mutandis, very similar to the previous one. The important

points to keep in mind are the following:

The parameters for a graph of negative Latin square type NL,(m) are v = m2,

k = g(m + 1), A = (g + l)(g + 2) - m - 2, p = g(g + 1).

0 The eigenvalues of this graph are r = g and s = g - m, with multiplicities (m + 1 ) (m - g - 1 ) and g(m + 1) = k respectively.

O The size and two possible intersection numbers for the corresponding two-intersection

set are N = g(m + l ) / ( q - l ) , A = g (q + m)/(q2 - q ) and B = (q(g - m) + m(g + 1 ) ( q 2 - q ) Again, we cal1 these numbers IVg7 A, and B, in order to emphasize

their dependence on g .

The properties of Al, A and B are now:

m NOW Rie need to prove that no hyperplane in PG(2n - 1: q) intersects SI U S2 in

B, + Bd points.

O The first three equations are now:

x + y = (m + l ) ( m - g - l ) / ( q - l ) ,

x + z = ( m + l ) ( m - g t - l ) / ( q - l ) ,

x + y + z + w = ( m 2 - l ) / ( q - 1 ) ;

the justification is the same as before.

The fourth condition is the same as before, namely

0 Finally, the solution of the system of equations is w = O (i.e., no hyperplone inter-

sects Si in Bi points and S2 in B2 points, as required), x = (m2 - ( g + gl)(m + 1 ) - l ) / ( q - 1 ) , y = g'(m + l ) / ( q - 1 ) and z = g(m + l ) / ( q - 1 ) .

These theorems have the following easy corollary:

Corollary 2. If Sl C S2 are both two-intersection sets i n P G ( 2 n - 1, q ) c o m p o n d i n g to

graphs of pseudo Latin (mpectively, negative Latin) square types L,(rn) and L#(m) (R-

spectively, ?VLg (m) and NLd (m)) then S2 - S1 is a two-intersection set c o m p o n d i n g to

a graph of pseudo Latin (respectively, negative Latin) square type Ld -, (m) (respectively,

-wf -9 (W-

Proof. Since it corresponds to the graph complementary to that determined b~ S2, the

set PG(2n - 1, q) - S2 is a two-intersection set of the same kind (pseudo or negative

Latin square) as Sl and S2. Now, the two-intersection sets SI and PG(2n - 1, q) - S2 are

disjoint and therefore either Theorem 11 or Theorem 12 applies. By noting that S2 - Sl

corresponds to the complementary graph of that defined by (PG(2n - 1, q) - S2) U SI,

we get the desired result. O

These theorems and corollary protide an interesting way of trying to obtain new

cyclotomic graphs from known ones.

The preceding discussion suggests the following question: under what conditions is

the union of two disjoint two-intersection sets a two-intersection set? We can Say the

following:

Theorem 13. Let S1 and S2 be two disjoint two-intersection seki in P G ( n , q ) , S1 havïng

size f i and intersection sizes Al > B1, and S2 having sire N2 and intersection stzes

-42 > B2. Then if Sl U S2 is a two-intersection set, one of the folloun'ng cases must hold:

1. The only two intersection sites of S1 u S2 are Al + A2 and B1 + B2 (2. e., every

hyperplane either intersects Si in Al points and S2 in A2 points, or intersects SI

in BI points and S2 i n BÎ points).

2. The nurnbers Al + B2 and BI + A2 difier, and the only two intersection sizes of

SI U S2 are Al + B2 and BI + A2 (i.e., every hyperplane either intersects S1 in Al

points and S2 i n B2 points, or intersects Si in Bi points and Sz i n A2 points).

3. Al + B2 = A2 + Bi, and either:

0 n o hyperplane intersects S1 and S2 in Al and A2 points, or

0 no hyperplane intersects SI and S2 in B1 and B2 points.

Proof. Suppose S1 U S2 is a two-intersection set.

If both Al + A2 and BI + B2 are intersection sizes of S1 u S2, then the other two

possibilities are forbidden (otherwise, since Al + A2 > Al + B2, A2 + BI > BI + B2,

S1 u S2 would have a t l e s t three intersection sizes).

If neither Al + Ag nor B1 + B2 are intersection sizes of S1 U S2, then necessarily the

tnro intersection sizes of S1 u S2 are Al + B2 and A2 + B I , which must be distinct.

Finally if only one of Al + A2, Bl + B2 is an intersection size of Si U S2, then the other

size must necessarily be one of Al + B2, Al +Bi. Assume, without ioss of generaiity, that

the intersection sizes are Al + A2 and .Al + B2. Then the hyperplanes that intersect S1

in B1 points cannot intersect S2 in B2 points (since B1 + B2 is not an intersection size of

Si U S2) - Thus, these hyperplanes intersect S2 in A2 points and SI U S2 in BI + -42 points-

It follows that BI + AÎ must equai either Al + A2 or Al + B2 (the only intersection sizes).

As B1 < Al, we must have B1 + A2 = .41 + Bz. O

We conjecture that the first two cases are impossible. As for the 1 s t case, pseudo

Latin and negative Latin square type graphs give examples of both situations.

2.3 Constructions

In t his section we present, without proofs, some of the known constructions of cyclotomic

graphs. In these constructions, 7 is a h e d element of q',, , and yL denotes the set

{x E IFqn : Tr(yx) = O). This set is a hyperplane (the proot of this is similar to the proof

of Theorem 9: yL when regarded as a set in the vector space is the kernel of a linear

functional, and as it is not identically O its dimension must be n - 1); moreover, since y1

and % determine the same hyperplane iff yl/y2 E 5 , the number of distinct hyperplanes

determined in this way is (Il$,, 1 / I l $ I = (qn - l ) / ( q - 1) and therefore every hyperplane

of PG(n - 1, q) c m be expressed in this fonn for some y E 5.. 1. (see 116, Thm. 9.11). Let n = 21, let z be a primitive element of Fqn, and let

R = (zq+l) be the subgroup of (q + 1)-th powers in q.. Let w = ~ ( q ~ ' - ~ ) / ~ ( q - ' ) , and

define the coset S of R by

[ R, if q ir even,

S = ( WR, if q is odd.

Then, if we cal1 c = (-l)', S is a two-intersection set with parameters ((qn - 1)/(q + 11, n, (9' - É) (9'-' +c)/(q+ l), (qu-l - 2 4 - ' - l) /(q+ 1)) (the size of the intersection

S n rL depends on whether 7 E S or not).

This construction gives, among ot hem, two-intersection sets and st rongly regular

graphs with the foiiowing parameters:

A l ~ l SRG

2. (see [16, Corollary 9.21). In the previous example, let A be a proper subset of

{O, 1,. . . , q } , let S = u ~ ~ ~ z ~ S , and let S-' = y 1 y-' E S. Then S is a two-

intersection set with parameters (1 AI (qn - l)/(q + 1), n, IA 1 (q' - e) (qf-' + e)/(q + l), (qu-l - 2 4 - ' - 1 + (]Al - l)(qL - c)(qL-1 + c ) ) / ( q + l ) ) , with the size of S n yL

depending on whether 7 E S-' or not.

3. (see 116, Example CY4, Section 91). Let IF# be a subfield of Fe, and let S be

the union of a t least two cosets of the subgroup Pd P# of $,, . This yields a two-

intersection set in the correspondhg projective geometry over IF'#. In particular,

if q' = q2 and 1 = 3 we obtain a two-intersection set with parameters (q2 + q + l ) t , 3, (q + l)t, (q + l ) t - q) in PG(2, q2), where 1 5 t 5 q2 - q.

4. (see [16, Section 71). Let S be a d-dimensional subspace of PG(n - 1, q) . Then S

is a two-intersection set with parameters (bd+', ni bd+', bd). These examples are not

very interesting, since the resulting strongly regular graph has p = O, Le., it is a

collection of disjoint cliques (the cliques in the affine region of PG(n, q) being the

affine points of the (d + 1)-dimensional subspaces of PG(n, q) which intersect the

hyperplane at infinity precisely in S).

Chapter 3

Computat ional Aspects

In this chapter we mention the algorithmic and cornputational issues we have to deal with

when searching for strongly regular graphs, finite geometries and related combinatorial

structures. Also, we explain some of the algorithms that we have developed in order to

solve sorne of the problems that arise in the search process.

3.1 Some common issues

Several of the problems that we face when searching for strongly reguiar graphs or finite

geometries are the same that a i s e in practically any combinatonal search. Some of the

most important are the following:

r should the search be exhaustive (i.e., comprise the whole search space) or heunstic

(i.e., explore only a selected part of the space)?

r shouid the search be deterministic or randorn?

0 given a newly generated object, is i t isornorphic to some other object already gen-

erated?

0 can the check for isomorphism be performed before the whole object is generated

(Le., is it possible to decide, early in the generation process, that the final result (s)

will be necessarily isomorphic to some already generated object (s) ?

In the particular case of Our study, we have restricted our search to those graphs

that can be found after pre-fusing the onginal matrix according to a subgroup of the

automorphisrn group of the matrix, with a sufficiently small number of orbits, and explore

these cases exhaustively in a detenninistic fashion.

With respect to the isornorphism detection problem, in general we have to wait until

the object is completely generated before checking for isomorphism: for example, the

algorithms that generate the possible extensions of a group output one complete object

at a tirne. One algonthm that might be modifiable to check for isomorphism of partial

solutions is the one for the fusion problem (see below). The idea of checking for isomor-

phism as partial solutions are found is the basic idea of the so-called orderly algorithms

for generation of combinatorial objects, see [47] and [51] for more details.

3.2 O t her computationai problems

In this section we briefly mention some other algorithmic problems that have axisen

during t his search process.

3.2.1 Graph isomorphism

Al1 throughout the search, we have had to decide whether two objects are isornorphic, or

mhether an object just generated had already been generated (in an isomorphic version).

We have had to do this for graphs, groups and matrices. Fortunately, al1 these objects

(and practically every combinatorial object we can imagine) can be represented as a

graph in such a way that the isomorphism structure is captured. More precisely: for

al1 these (and many other) classes of objects, it is possible to define a map f from

the set of objects of the class into the set of graphs such that two objects O and 0' are isomorphic if and only if the graphs f (O) and f (0') are isomorphic. The problem of

checking isomorphism is well solved in practice, if not in theory: although it is not known

whether GRAPH ISOMORPHISM E P, there is at least one program, nauty [46] that solves

this problem very efficiently in most cases. There have been several instances where the

program has taken hours (or even days) to complete its task, however, we believe that

it is possible to improve its performance by fine-tuning some of the parameters given to

the search. In any case: nauty is the tool of choice for al1 the isomorphism checking (be

it of graphs, groups, matrices, or anything) that we need.

3.2.2 Some ubiquitous N P-hard problems

We mentioned above that it was not known whether GRAPH ISOMORPHISM E P. Also,

it is not known either whether it is an NP-complete problem. But there are other NP-

complete problems that appear (disguised as their NP-hard enumeration or optimization

versions), specially in the analysis stage after al1 the graphs have been generated. Several

of the properties of the graphs that are interesting to us (what size the maximum cliques

are, hou7 many of them are there, can the set of edges be partitioned into cliques) require

us to solve an NP-hard problem. Again, we are fortunate that these problems can

be solved efficiently in practice, for the cases that we are studying. Of course, unless

P = NP, for any algorithm that solves problems such as Clique Finding or Sei Partition,

it would be possible to find a family of instances that would force the algonthm to take

more than polynomial time to solve them. For Clique Finding we use our own very simple

algorithm, and for Set Paztition we use Mathon's fast set-partitioning algorithm [44].

3.2.3 Computing in a projective geometry

Another class of problems for which we need efficient algorithms is that related to geo-

metric computations in a projective geometry. Some of the important problems are:

given two points, what other points are in the line that joins them?

more generally, given d + 1 independent points, what other points are in the d-

dimensional space spanned by t hem?

O given two subspaces, how can we compute their intersection in a fast way?

given two points in the affine region of PG(n, q), what is the point a t infinity of

the line that joins them?

given a point and a d-dimensional subspace, is the point in the subspace?

m more generally: given a d'-dimensional subspace and a d-dimensional subspace

(d' < d), is the d'-dimensional subspace included in the d-dimensional one?

We have developed algorithms for these and other related problems, based on the repre-

sentation of the points of PG(n, q) as vectors on q+' (homogeneous coordinates). The

key idea for these algonthms has been to represent each d-dimensional subspace by a

canonical basis (set of d + 1 points that span the subspace). We outline the definition of

this basis as follows:

0 each point P E PG(n, q) is represented in homogeneous coordinates as a vector

formed by n + 1 elements of Fq, the last non-zero of which is 1;

0 for each point P E PG(n, q) we define its level to be the number of its last non-zero

coordinate;

r we select one subspace of each dimension in PG(n, q), and denote them So, Si , . . . , Sn =

PG(n, q ) , as follows: Si is the subspace formed by al1 the elements of levels between

O and i, clearly Sa E S1 c - - - Sn = PG(n,q);

r given S a subspace of PG(n, q), let r be the smallest number such that S C Sr but

5' $Z Sr-1;

if we let S' denote S n Sr-1, then S can be expressed as (S' + P), where P is any

point in S - Sr,I;

the rank formula tells us that S' has dimension d - 1;

r let Po, Pl , . . . , Pd- 1 be a canonical basis of S' .

r let each point Pi (O 5 i 5 d - 1) have level li, there is only one point - cal1 it Pd

- in S - Sr-1 which has a O in every coordinate corresponding to some l i ;

the canonical bais for S is {Po, Pl, . . . , Pd-1, P d } .

Given any generator of a subspace S of PG(n, q), a canonical basis for S can be found

as follows:

A LGORITHM 3.1 : CANONICAL BASIS

Input: a generator {Qo, QI,. . . , Q d ) of a subspace S of PG(n,q).

Output: a canonical basis {Po, Pl,. . . , P l ) of S (1 5 d is the dimension of S).

begin

D First part: ensure that the points Qi are sorted by level and that there is at most one such

point at each level.

for i from d downto 1 do

for j from O to i do

if the level of Q, is higher than the level of Qi then

swap Qi and Q,.

if the level of Qj is the same as that of Qi then

Q t Qi - Q j ; if necessary, normalize Q j-

end if

end for

end for

D Second part: for each point in the set, make O aii the coordinates corresponding to other

points' levels.

for i fkom O to d d o

Let f i be the level of Qi-

end for

for i fkom O to d do

Pi = Qi

for j from i - 1 downto O do

Let z be the coordinate at level 1, of Pi.

D By definition of level, the coordinate at level lj of Pi is 1.

P i t P i - xPj .

if Pi = O then delete Pi from the list.

end for

end for

return (Po, Pi,. . . ,Pd) .

end

The assignment Pi c Pi - xPj has the effect of tuming the 1,-th coordinate of Pi into

O; as P, has only zeros in al1 the coordinates with index > lj? this assignment does not

disturb the values of the coordinates lj+i, l,+*, . . . , I i - i of Pi? which have been set to O in

previous iterations of the loop. If Pi becomes the zero vector a t any point, that means

that Pi can be expressed as a linear combination of other points in the generator and

therefore is inessential for the basis.

It is not hard to see that the complexity of this algorithm is O(nh) (both parts have

O(@) iterations, each with an inner operation that takes time O(n): either Q, + Qi - Q, O, Pi c Pi - x P j ) .

We illustrate these ideas with an example: in PG(5,3), consider the two-dimensionai

subspace spanned by the set of points (120021,22101 1,020211); we will find the canonical

basis for it by applying the above algorithm. During the first part of the algorithm, the

points Qi change as follows:

Qo QI Q2

120021 22101 1 020211 Initial points

200220 221011 020211 Qo t Q2 - QO

100110 221011 020211 Normaiize Qo

100110 102200 020211 Q l t Q2 - Q I

100110 201100 020211 NorrnalizeQL

201100 100110 020211 Swap Qo and QI

During the second part of the algorithm. we get the following values for Po, Pl and Pz:

The attentive reader will realize that this method is essentially Gaussian elimination

applied to the matrix of coordinates of the original basis.

Using canonical bases, it is possible to sort the subspaces of a given dimension in a

uniquely defined order: for each subspace we consider its canonical basis, sorted by the

level of its elements; the order between subspaces is given by the lexicographic order of

the coordinates of the elements of their bases. In this way, we can refer to "the 100-th

3-dimensional space7' or the "245-th hyperplane7' in a given projective geometry.

Canonical bases allow us to easily solve some of the algorithmic questions posed above;

here ive sketch the algorithms for each of them.

Projec t ion to idnity Given two points Ql and Q2 in the affine region of PG(n, q),

what is the point a t infinity in the line that joins them?

The answer to this question can be obtained by considering the vectors of hc+

mogeneous coordinates that represent Q and Q2, subtracting bot h vectors, and

normalizing the resulting vector (Le., dividing throughout by the last non-zero co-

ordinate). We will cal1 this point the projection at anjinity of the line QI + Qz.

The complexity of this algonthm is O(n).

Incidence Given a point Q and a subspace S with canonical basis (sorted by level)

{Po,. . . , Pd}, is Q a point of S ?

The obvious way to answer this is to compute a basis for the subspace generated

by {Po, . . . , Pd, Q). In this way, Q E S iff the canonical basis output by the above

algorithm is exactly {Po, - . . , Pd, Q); as mentioned above, this would take time

O(ndz). However, a more efficient aigorithm can be designed:

Let 1 = Level(Pd). If Level(Q) > 1 then clearly Q $ S (for, S is totally contained in

PG(I, q) while Q is not in PG(1, q)). If Level(Q) < 1 then, if Q E S, any expression

of Q as a linear combination of {Po, . . . , Pd} must contain Pd with coefficient 0' and

therefore in this case the problem is reduced to decide whether Q is in the subspace

spanned by {Po,. . . , Pd-1).

It only remains to analyze the case when Level(Q) = Level(Pd). If Q = Pd then

trivially Q E S; otherwise Q E S iff the projection at infinity of the line Q + Pd is

in the subspace generated by {Po,. -. , Pd-l}-

The total complexity of this algorithm is O(nd), since at most d + 1 iterations will

be needed, each needing a t most one "projection a t infinity" operation.

Inclusion Given two subspaces S1 and S2 of PG(n,q), the easiest way to determine

whether SI C S2 is to apply the above incidence algorithm to determine whether

each of the points making up the canonical basis of SI is an element of S2. If the

dimensions of Sl and S2 are dl and d2, respectively, the complexity of this algorithm

is 0(ndld2).

Intersection Given two subspaces S1 and S2 of PG(n,q), the easiest way to determine

SI n S2 is the following:

Let the dimensions of SI and S2 be d and 1 respectively. Let {Po,. . . , Pd) and

{Qo, . . . , Ql) be the canonical bases for SI and S2, and let Si and Si denote

the subspaces spanned by {Po, . . . , Pd-l ) and {Qo, . . . , Q1-l) respectively. If

Level(Pd) # Level(Q1) then Sl n S2 = Si n S; (for, no point in the affine part

of S1 is in S2, and vice versa, since al1 the affine points in S1 have the same

level as Pd and ail the affine points in S2 have the same level as Ql).

On the other hand, if Level(Pd) = Level(Q1) then let R be the projection at

infinity of the line Pd + Q1. If R E Si n Si then SI n S2 = ((Si n Si) + Pd) =

CHAPTER 3. COMPUTATIONAL ASPECTS

((Si n Si) + QI) , otherwise Si n S2 = Si n Si.

3.3 The main algorithrns

In this section we present the algorithrns that form the core of our methodology to search

for strongly regular graphs. These aigorithm are:

1. an algorithm to find the eigenvalue matrix of an association scheme;

2. an algorithm to generate the possible matrices that can be used for fusing;

3. an integer-programming based algorithm to quickly decide whether a given matrix

leads to at least one solution;

4. an algorithm to find, given the eigenvalue matrix of a scheme, al1 the sets of columns

whose fusion leads to a strongly regular graph with the desired parameters.

Although in the examples presented here we have had no need to use the first algorithm

(since the eigenvalues of the cyclotomic schemes are easy to cornpute), this algorithm

can be important for other examples where a closed formula for the eigenvalues does not

exist.

As for the next two algorithms, the need for them stems from the large size of the

matrices and the large number of groups we are working with: as it is infeasible to apply

the exhaustive search algorithm to matrices with more than 300 columns - for instance,

the eigenmatrices of the cyclotomic schemes C(1024,3) and C(729,2) - we do not apply

it directly to the eigenmatrix of the scheme; instead, we use the second algorithm to

determine groups of automorphisms of the mat* that will yield smaller matrices after

fusing columns according to their orbits. Now, the number of such smaller matrices is

large - usually in the hundreds or thousands - and only a few of these yield solutions;

therefore, applying exhaustive search to al1 these small matrices would be a waste of

time, and here is where the third algorithm cornes into play. Once the small matrices

that do yield solutions have been determined, the fourth algonthm - exhaustive search

- is applied to each of them.

3.3.1 An algorithm to find the eigenvalue matrix of an associ-

ation scherne

Consider a d-class association scheme on a set X. We know (see [12][Thm. 3.31) that

the eigenvalues of the adjacency matnces Al,. . . , Ad of this scheme are the same as the

eigenvalues of the intersection matrices L1, . . . , Ldy and that these matrices also have a

common set of eigenvectors. As these matrices are much smaller than the corresponding

adjacency matrices, we prefer to work with them rather than with the latter.

In order to find a basis of Hfd formed by common eigenvectors of the matrices L I , . . . , Ld,

ive need to find the minimal idempotents of these matrices, Le., the subspaces of Rd which

are invariant under these matrices. As each of these idempotents is the intersection of

one eigenspace for each of these matrices (namely, the only eigenspace of that matrix

that contains the given idempotent), we must express the whole space as the direct

sum of al1 possible intersections of eigenspaces of the matrices Li. This is achieved as

follows:

ALGORITHM 3.2: MINIMAL IDEMPOTENTS

Input: the d x d intersection matrices LI, . . . , Ld. Output: a List of subspaces that span Wd and are eigenspaces with exactly one eigenvalue for

each matrix Li,. . . , Ld.

begin

Initialize a set S as the empty set

Find the eigenspaces of the matrix Li, and add each of them to S.

for i ikom 2 to d

If aU the eigenspaces in S have dimension 1, return S

else find the eigenspaces Ul, . . . Ut of Li, and replace each space

Wj E S of dimension > 1 with the non-empw intersections LJk n Wj. end for

return S

end

Once we have expressed Rd as a direct sum of the elements of S, as each matrix Li has

exactly one eigenvalue for each of these subspaces, it is easy to compute the eigenvalue

matrix. The key sub-algorithm is the one used to compute the intersection of two vector

subspaces; for this we apply the method that appears in [32, pp. 429-4301.

3.3.2 An algorithm to find smaller matrices

-4s was mentioned in Section 2.1, the fusion problem is hard to solve for schemes with

many classes. The exhaustive search solution that we have implemented works fast

enough (about a couple of hours of CPU) for schemes with at most 40 or 50 classes (in

some instances it can work fast for even more than that, but that is rare).

What can be done if we want to find strongly regular graphs by fusing classes in

association schemes with more than 100 or 200 classes?

One way to make such an instance of the problem tractable (at the expense of possibly

missing some of the solutions) is to assume that some of the classes will always go together,

and to apply Our exhaustive search method to the resulting matrix. In a sense, we d l

be prescribing a "pre-fusion" of the classes, and this will allow us to work with a smaller

matrix.

A natural way of pre-fusing classes is by taking some small subgroup of the auto-

morphism group of the eigenvalue matrk, and fuse the classes according to the orbits

determined by that subgroup on the set of columns.

In order to systematicdly explore al1 possibilities, we would like to have a t Our disposa1

al1 the subgroups of the automorphism group of the eigenvalue matrix, and to select from

among these those that determine a small enough number of orbits in the columns as to

make an exhaustive search feasible. However, because of the large number of subgroups,

this approach is impractical (such programs as GAP [28] allow the user to compute the

whole lattice of subgroups, but for the examples that we want to tackle the time and

space requirements are prohibitive).

We have devised instead a procedure that will give us, if not every possible subgroup

of a a given permutation group G, a t least a large portion of them. Before presenting

this procedure, we will give the following

Definition 11. Let H be a subgroup of a pemutation group G acting on a set X . Let g

be an element of G - H . We wàll say that g respects the orbits of H if the image under

g of an y orbit of H is again an orbit of H .

In group theoretic terrns, the orbits of H are sets of imprimitivity of the group

( H U {9H-

The basic idea for Our procedure is as follows: suppose that H is a subgroup of G,

and let its order be n. Let the factonzation of n into (not necessarily distinct) primes be

n = pl132 - . . p k . Then we can find a sequence of k elements gl, 92, . . - , gk of H such that

for each i, 1 5 i 5 k, the order of gi is pi and the order of the subgroup i& generated by

{gi ,g2, . - . , g,) is plpl . . . pi (we define Ho t o be the trivial group (1)). We will consider

al1 the sequences of elements h l , h2 , . . . , hk of orders pi, a,. . . ,pl, where each element hi

respects the orbits of the group Hi-t . At each step, we will discard those elements that

determine a set of orbits that would yield matrices isomorphic to those already obtained

(since from that point on, the possible extensions of the corresponding groups would

yieId isomorphic matrices).

In this way, we determine a lattice of matrices, each of them labeled by a subgroup

of the group of automorphisms of the original matrix; each rnatrix is determined by the

orbits of the group it has as label. This lattice has at the root the original matrix, with

label Ho = {l), and the successors of each node are the matrices determined by groups

resulting from expanding the current label by each of those prime-order elements in G

that respect its orbits.

How to determine, then, precisely those prime order elements that yield non-isomorphic

matrices? The answer is partially given by the foliowing daim:

Claim 11. Let H be a subgroup of the automorphism group G of the eigenvalue matriz

P of an association scherne, and let' GH be the group of al1 elements of G that respect

the orbits of H . Then given two elements g,gl of G - H , the matrices obtained when

pre-fusing P according to ( H U g) and (H u g') are isomorphic i f g and g' are conjugate

in GH.

Proof. That elements of G conjugate in G (or in the symmetric group, for that matter)

yield isomorphic matrices is clear, since the correspondence between the entries of the

matrices is given by the correspondence between the orbits of the groups. O

IVe conjecture that the converse of the previous claim also holds; we have not been

able to prove it, but even if it were true, we do not really have the chance to use it, see

the next paragraph.

From a practical point of view, we must mention that finding the conjugacy classes

of elements of GH is in most cases outside the realm of feasibility; this is why instead

ive settle for finding the conjugacy classes in Sylow subgroups of GH (this has the effect

of not recognizing two elements g and g' as conjugates if the element h E G H such that

'This notation is somehow unfortunate, but unambiguous: it cannot be confusecl with a pointwise stabilizer since H is a subgroup of G, not a subset of X.

g = hg'h-l has a non-prime order, or a prime order different to that of g and gr) and

t hen separately rejecting isomorphic matrices.

In summary, the algorithm to find the lattice of matrices is as follows:

ALGORITHM 3.3: LATTICE OF MATRICES Input: The eigenvaiue matrix P of the scheme.

Output: The lattice of matrices that can be used for fusing.

1. Initially, set H = (1) and set the lattice as empty.

2. Fuse the rows of P according to the orbits induced by H .

3. Add t bis matrix to the lattice, as a successor of the matrix whose label is extended by

H .

4. if the resulting matrix is smaii enough then

5. go back to the other tasks that are pending, if any.

6 . else

7. Let GH be the subgroup of the automorphism group of P that "respectsn the orbits

of H.

8. For each prime ?r dividing IGH 1, let S, be a Sylow T-subgroup of GH.

9. Find (up to conjugation) al1 different elements g of order .rr in S,, and for each of

t hem consider the group Hg = ( H U g).

10. Many of the groups Hg produce matrices isomorphic to others already generated

when used to fuse the rows of P. Reject all those groups Hg yielding isomorphic copies.

11. For each of the surviving elements g, go to step 2 with Hg instead of H.

12. end if

13. Return the lattice as calculateci

3.3.3 An integer programming formulation for the fusion prob-

lem

Here we formulate* an integer programming version of the FUSION-E problem. This is

useful because we can usually (using CPLEX or some other integer programming pack-

age) check the feasibility of this IP problem much faster than running our exhaustive

search algorithm, and thus we can Save a substantial amount of CPU if we know before-

hand that the search will be fruitless. In fact, if it were possible to perform a complete

enumeration of the feasibie solutions of an IP problem using CPLEX or some other simi-

2This formulation was suggested by Lucia Moura.

larly efficient software system, the algorithm descrîbed earlier might be rendered useless;

however, we are not aware of a program capable of doing this a t the moment.

Let the original association scheme have d classes, let P be its eigenvalue m a t r k

(thus, P is a (d + 1) x (d + 1) mat*), and let the parameters of the desired strongly

regular graph be (v, k, A, p) and let r and s be the eigenvalues of such a graph. We will

delete the first column of P (since in any scheme it will consist entirely of 1's). Thus we

are Ieft with a (d + 1) x d matrix, and we need to select a set of columns in such a way

that in the first row, the selected columns must add up to k and in al1 the other rows,

they must add to either r or S. If we number the r o m of P from O to d and its columns

from 1 to d, the formulation is as follows:

I/ariab[es: For each column P , of the matrix P , there will be a binary variable xj. For

each row Pi. of P, there will be a binary variable yi.

Constraints: For each row Pi. of P (with i > O), there will be a constraint

Corresponding to the @th row of P , there will be a constraint

Cost function: .4ny cost function will do.

The meaning of these variables and constraints is as follows:

A variable x, will be 1 if the corresponding column is "selected", O otherwise.

A variable Yi will be 1 if the surn of the entries of the selected columns in the i-th

row is r , otherwise (Le., if said sum is s), it will be O.

The constraint xj PWxj = k means that the surn of the valencies of the selected

columns (classes in the scheme) must equal the valency of the graph.

The constraint x, Pijxj + ( r - s)yi = r means that the sum of the entries of the

selected columns in each row except the first rnust equal either r (if Yi = 1) or s (if

Yi = O).

Of course, as we are interested only in the feasibility of the problem, it is not important

which cost function is used (although different cost functions might have an effect in the

speed with which a solution is found or infeasibility is detected).

-4 particular integer programming package that we have used, CPLEX, can solve

the above mode1 very efficiently: for matrices of about 80 rows, it usually takes only a

few minutes to determine whether a subset of their rows determines a strongly regular

grapti with a given set of parameters. The only drawback of this approach is that, if the

problem is determined to be feasible, we have found only one solution. One goal for future

developrnent is to integrate the feasibility checks provided by the integer programming

method with the general fusion algorithm described above (3.3.4).

3.3.4 An exhaustive search algorithm for the fusion problem

Let 1b.f be the eigenvalue matrix of an association scheme on v vertices, and let (v, k, A, p)

be a feasible set of parameters for a strongly regular graph on u vertices. Let r and s

be the corresponding eigenvalues. We would like to know if there is a subset S of the

columns of M such that, for every row of M, the sum of the entries in coIumns of S is

either T or S.

The basic idea behind Our algorithm is that of backtracking: we will systematically

explore al1 possible combinations of columns. However, there are several ways in which

we improve the efficiency of such an algorithm:

1. We perform a separate analysis for each possible cardinality of S. This allows us

to assume that cardinality as known, see the next item for an example.

2. Before we start the backtracking process (where the cardinality of S has been fixed),

we find for each row what combinations of its entry values can be used in order to

add up to the desired eigenvalues. Thus, for example, if a row has 20 entries having

value 5 and 8 entries having value -4, the desired eigenvalues are 3 and -6 and

we are assuming that the cardinality of S is 6, then the only possibilities for this

row are either three columns with 5 and three with -4, or two columns with 5 and

four with -4. The total number of possible combinations that we should check for 20 8 this row is ( , ) (,) + ( y ) (0 = 1140 56 + 190 70 = 63840 + 13300 = 77140, while

a brute force enurneration of al1 the 6-sets of columns would require to consider

( y ) = 376740 such sets.

After we do this for each row, we choose one row for which the total number of

combinations to examine (in principle) is minimum, and set out to explore only the

possible "goodn combinations of columns for that row.

3. The analysis mentioned in the previous item does not only help choose a "base row"

that will guide the main backtracking process; it also gives us some information that

can be used dunng the backtracking to speed up the search. For instance, suppose

the row mentioned above is not the one chosen at the end (there is a better row).

Still, we know that no more than 3 columns with entry 5 in this row can be selected.

If at any stage during the backtracking process we detect 4 columns with entry 5,

n7e can abort the search and proceed to the next candidate.

Of course, keeping track of the information needed to perform this test has its cost,

and therefore a compromise must be reached: these tests pay off only if they are

v e l likely to trigger the decision to abort the search at an early stage. A very

crude estimate of this likelihood can be obtained by assuming that the colurnns in

the test row are chosen at random, and computing the expected number of columns

that need to be chosen so that the test fails (in our case, we should estimate how

many columns must be chosen so that four columns with entry 5 are chosen). If

this number is too close to the assumed size of S, the test is not worth the effort.

In summary, a high-level description of the aigorithm is as follows:

ALGORITHM 3.4: FUSE Input: A matrix M, two desired eigenvalues r and S.

Output: the sets S of coltimns that give only the desired eigenvalues wben added up.

Find the possible sizes kmin,. . . , km, of S.

for k from Ccmin to km, do

for each row, find the combinations of values that add up to r or S.

Select the row for which the number of combinations to try (in principle) is minimum.

For al1 the other rows, find al1 possible "testsn and their likelifioods.

Sort the tests according to likelihood, discard the least likely ones.

For the selected row, try al1 possible combinations of coliimns that would add up

to r or s, while keeping track of information needed for each test. At each step,

try the tests to decide whether the current search path should be aborted.

end for

Other ideas that could be considered in order to make the process more efficient are

the following:

a Incorporate information about the isomorphism group of the matriu. This would

allow us to consider one column as a representative of eacb orbit, greatly reducing

the total time needed. In addition, each time a column is assumed to be fixed, the

tests descri bed above are strengt hened.

Incorporate the techniques from integer prograrnming to the search: such methods

as "branch and bound" or "branch and cut" might reduce the search time.

Chapter 4

Computat ional results

In this chapter, we present the raw results from the computational searches that we have

performed searching for strongly regular graphs that can be obtained by fusing classes

in the cyclotomic schemes on 243, 256, 625, 729 and 1024 elements. In al1 but the last

case, the scheme considered has been of the form Chn, p - 1) (as discussed in Chapter

1.9.5). For the case of 1024 elements, we have focused on C(1024,3) instead of C(lO24, l ) ,

since the tremendous size of the automorphism group of the latter scheme would render

our method useless (as a way of cornparison, the sizes of the automorphism groups of

C(1024,3) and C(1024,i) are 516984510873600 and 366440137299948128422802227200

respectively). One way of interpreting the effect of this decision is that, for the case of

1024 vertices, we are working in the projective geometry PG(4,4) instead of PG(9,2);

however, once we have found a graph we can interpret it as a point set in either of these

projective geometries.

In the first section we explain the structure of the lattice of matrices introduced in

3.3.2.

In the second section we show, for each feasible parameter set for a strongly regular

graph on some of the numbers of vertices that we study (Le., 243, 256,625, 729 and 1024)

for which we successfully applied our methodology, selected information such as: whether

graphs with those parameters were known or not, the parameters of the corresponding

two-intersection set and two-weight code, how many groups in the corresponding lattice

yielded a solution, and finally how many non-isomorphic graphs wit h those parameters

we have found.

4.1 The lattice of matrices

In this section we explain the structure of the lattice of matrices associated to an as-

sociation scheme C(qn, q - 1). These lattices are generated by applying the algorithm

presented in 3.3.2. We have set an arbitrary cutoff of 40 orbits to be the maximum

number of orbits for a group that is not extended; in two cases - see below - we have

been unable to completely generate the lattice up to this cutoff. In those cases, we have

settled for generating a significant portion of the lattice, discarding nodes in the deeper

levels of it.

As an example, we first show al1 the non-isomorphic matrices in the lattice corre-

sponding to the case of 625 elements; there are 84 of these matrices in total. We have

chosen this example as illustration since it has the smallest lattice of al1 the cases that

we have analyzed.

We adopt the following naming convention for the matrices listed in this table: the

group corresponding to the matrix a t the root of the lattice - its label - is designated

by Ho, and in generd, if a matrix - considered as a node of the lattice - is a successor

of another, and is obtained from this one by considering an additional generator of prime

order T , which occupies position i in the list of possible extensions of order rr, then the

designation of the group which will label the successor is obtained by adding the string

ri to the designation of the parent. We follow this convention, with one exception: the

root is designated by Ho, but we designate its successors as if the root were designated

H (Le., instead of Ho20 we mi te simply H20, etc.).

Xfter we show the lattice of matrices for the case of 625 vertices, tve present a summary

of t h e number of matrices found a t each level in each of the lattices.

4.1.1 Exarnple: the cyclotomic scheme on 625 vertices

We now show the lattice of matrices corresponding to the cyclotomic scheme on 625

verticcs. Here we are pruning the lattice in order not to show isomorphic branches; any

missing labels, such as H2&!0 or H202221r correspond to matrices isomorphic to some

other matrix in the lattice. The foilowing table shows, for each matrix M in the lattice:

its label LABEL(M), which is a group whose orbits induce the matrix;

r, the size of its automorphism group, which we denote by GM;

0 its size 1 MI, Le., its number of columns; this equals the size of the set O ( H ) of

orbits determined by H .

As we are not showing the different (isomorphic) ways to obtain each matrix, the

lattice takes the appearance of a tree; the matrices in the following table are listeci in

what would be a pre-order if it were a tree (i-e., a t each step we list a node before listing

its children, or as we said above, its successors).

Table 4.1: Lattice of matrices for C(625,4)

The following picture shows this lattice:

CHAPTER 4. COMPUTATIONAL RESULTS

Figure 4.1: The lattice of matrices for C(625,4).

CHAPTER 4. COMPUTATIONAL RESULTS 65

The total number of matrices found at each level of each lattice is given by the

following table:

Table 4.2: Number of matrices at each level of the lattice

Level

-4 fem explanations about this table are in order. In principle, the idea was to apply

the algorithm described in 3.3.2 to completion, Le., until al1 the unextended nodes in the

lattice had no more than 40 orbits. This could be done for the orders 243, 625 and 729.

In the cases of 256 and 1024, the number of groups already obtained a t level 3 or 4 was

so big that the space requirements made it impossible to continue. In the case of 256

vertices, we settled for completing the first 4 levels of the lattice; in the case of 1024 we

completed the first 4 levels and extended only those groups at level 3 with more than

100 orbits.

4.2 The graphs found

The following table gives, for some selected parameter sets for which the methodology

is successful, some selected information. In the first column we give the parameters

( u , k: A, p ) of the strongly regular graph. The second column gives the parameters of

(N, n, A, B) of the corresponding projective two-intersection set. The third column gives

the parameters (n, k, wl, w2) of the corresponding projective two-weight [n, k] code with

weights wl and wq. The fourth column contains a comment: here we indicate whether

these parameters correspond to graphs of negative Latin square type, and whether graphs

with these parameters are new. The fifth column tells us how many different groups

yield graphs with those parameters. Finally, the sixth column shows us how many non-

isornorphic graphs with those parameters were found; in al1 the cases but one (for which it

is known that the graph is unique), we indicate 2 since we have not necessarily exhausted

al1 the graphs with those parameters.

CHAPTER 4. COMPUTATIONAL RESULTS

Table 4.3: Information about the graphs found

SRG 2-int. set

(111 5,5,2)

(55,5,19,10)

(51,8,27,19)

(68,8,36,28)

(85,8,45,37)

(102,8,54,46)

(119,8,63,55)

(26, 41 61 1)

(3% 4,9,4)

(521 4, 12, 7)

(65,4,15,10)

(56,6,201 11)

(84,6,30,21)

(98,6,35,26)

(112,6,40,31)

(126,6,45,36)

(140,6,50,41)

(154,6,55,46)

(168,6,60,51)

(77,5,21,13)

(99,5,27,19)

(1 10,5,30,22)

(121,5,33,25)

(132,5,36,28)

(143,5,39,31)

(154,5,42,34)

(165,5,45,37)

Code parameters

(11,5,6,9)

(55,5,36,45)

(51,8,24,32)

(68,8, 32140)

(85,8,40,48)

(102,8,48,56)

(1 l9,8,56,64)

(26,4,20,25)

(39,4,30,35)

(52,4,40,45)

(65,4,50,55)

(56,6,36, 45)

(8496,547 63)

(98,6,63,72)

(1 12161721 81)

(126,6,81,90)

(14076, !Ml 99)

(154,6,99,108)

(168,6,108,117)

(77,5,56,64)

(99,5,72,80)

(1 10,5,80,88)

(121,5, 88,96)

(132,5,96,104)

(143,5,104,112)

(154,5,112,120)

(165,5,120,128)

- - -

Comment

It must be kept in mind that in al1 the cases not al1 the groups were tried to completion:

in most cases we tried only those inducing a t most 40 orbits, but in some cases, namely

those where there were more than a hundred groups, we tried only a few small groups

(the reason for this is that when there are many groups yielding solutions we can consider

t hat case as "easy" , and therefore not as interesting as other cases). In any case, a11 the

groups with no more than 100 orbits were examined using CPLEX and this provided a t

CHAPTER 4. COMPUTATIONAL RESULTS 67

Ieast one solution for each such group yielding solutions (this solution may or may not

have been found when searching a t deeper levels of the lattice).

In general, a graph will be found in the matrices induced by any subgroup of its auto-

morphism group, and only in those matrices. Thus, a graph with a large automorphism

group (and many different subgroups) will appear many times in different matrices, while

a more rigid graph (smaller group) wili not appear as often.

Another consequence of the property presented above is that, given a group G, its

size is a lower bound on the sizes of the automorphism groups of the graphs that can be

found in the matrix induced by G. It is interesting to note that in most cases, the size of

the automorphisrn group of the graph is larger than the sizes of the groups which were

actually used to find it; this means that that graph could have been found even if we

were searching deeper in the lattice (i.e., assurning a larger group) than we actually did.

On the other hand, there are some graphs for which this is not true. For instance, the

only two graphs found with parameters (1024,297,76,90) are the solutions found with

CPLEX for two groups of size 18, one inducing 65 orbits and the other 76. These two

graphs have as automorphism groups precisely the groups for which they were found.

Chapter 5

Analysis of some of the graphs found

In this chapter we study some interesting properties of the graphs found by the process

described in the previous chapters.

5.1 Cliques and subspaces

The first property we are interested in is the size of the cliques of the graphs; in al1 the

graphs found ive can find cliques related to the subspaces of the projective geometries in

mhich the graphs are embedded. We will fix the notation as follows: frorn now on, we

will assume that our graph is deterrnined by a two-intersection set S c H, where H is a

copy of PG(n - 1, q) taken as a hyperplane at infinity in PG(n, q) (and thus the graph

has as vertices the affine points of PG(n, q)). In the graph we might find cliques of size

q, which are affine Iines in PG(n, q), and are determined by points in S; q2 (affine planes

in PG(n, q): corresponding to lines totally contained in S), and so on. In general, if a

d-dimensional subspace S of H is completely contained in S, it determines qn-d-l cliques

of size qd+l in the graph, namely al1 the (d + 1)-dimensional subspaces of PG(n, q) which

contain S' and are not completely contained in the H . These cliques of size qd+l form a

spread of the graph, i.e., they partition its vertex set.

Other cliques of the graph are formed as follows: consider a set A of subspaces, each

of which is completely is completely contained in S, and let P be the set of d l points

contained in some element of A. Suppose that for every pair of points x, y E P there is at

least one subspace in A containing both x and y (if the number of subspaces containing

the two points is the sarne for every pair of points then the subspaces form a block design,

but this is not necessary). Then, given any point v in the affine part of PG(n, q ) , we can

form a clique of size v(q - 1) + 1 in the graph, by taking al1 the lines that join v to points

of P.

5.2 Geometries

A natural question to ask about the cliques in a strongly regular graph is: can we find a

partition of the set of edges into cliques of the same size? If the graph is pseudo-geometnc,

corresponding to a pg(s, t, a), then finding a partition of the edges into maximum cliques

is equivalent to finding a geometry. If the parameters corresponds to a semipartial ge-

ometm the maximum cliques may not be of the right size for the geometry and thus a

partition of the edge size into smaller cliques should be sought.

For seven parameter sets, we were able to find such a partition:

1. The only graph found with parameters (243,22,1,2) supports a spg(2,10,1,2).

This graph is known to be unique;

2. The ouly graph that we have found with parameters (256,51,2,12) supports a

spg(3,16,1,12), a partial quadrangle found by Cameron;

3. One of the graphs with parameters srg(256,119,54,56) supports a spg(7,16,4,56),

this geometry is due to Thas;

4. The only graph that we found with parameters (625,104,3,20) supports a spg(4,25,1,20),

found by Cameron;

5. The only graph that we have found with parameters (729,112,1,20) supports a

spg(2,55,1,20); this geometry was already knoam. This graph is also known to be

unique;

6. One of the graphs with parameters (729,168,27,42j supports a partial geometry

pg(8,20,2); this same geometry had been found in 1997 by Mathon [45].

7- Each of the new graphs srg(625,156,29,42) admits a partition of the edges into

maximum cliques, yielding a geometric structure similar to a semipartial geometry,

but where the number of points in a line collinear to a point outside the line can be

either O, 1 or 2 (i.e., can take three values, unlike the case of semipartial geometries

where it can take only two).

CHAPTER 5. ANALYSIS OF SOME OF T H E GRAPHS FOUND 70

There are three other parameter sets corresponding to graphs of negative Latin square

type for which a possible geometry could exist; so far these geometries are unknown:

srg(1024,231, 38,56), with a possible spg(7,32,2,56);

srg(1024,330,98,110), with a possible pg(15,21: 5 ) .

0 srg(1024,363,122,132), with a possible spg(3,120,2,132).

For the first two cases we performed an additional search of two-intersection sets in

PG (9 ,2) , considering some selected subgroups of the corresponding automorphism group.

In the case of srg(1024,231,38,56), the additional search yielded 19 graphs (some of

them already found in PG(4,4)); of al1 the graphs found only three had enough cliques

of size 8; however, in two cases those cliques did not partition the set of edges, and in

the other they did not even cover it.

In the case of srg(1024,330,98,110), the additional search yielded 44 graphs; over al1

the graphs found only two graphs had the right number of cliques of size 16, but in none

of these did the cliques cover the set of edges.

In the case of srg(1024,363,122,132), we checked whether the lines of PG(5,4) inter-

secting S (the two-intersection set) in exactly one point gave a linear representation of a

s p g ( 4 : 121,2,132); this is not the case for any of the graphs found. For this to be the case,

S must be what is called an spg-regulus, Le., every line of PG(4,4) must intersect S in

either 1 or 3 points (in which cases the lines are called tangents and secants respectively),

and each point in PG(4,4) - S must be in the same number of tangents. None of the

graphs found satisfies this condition.

5.3 Spreads in geometries

-4s we mentioned before, the set of affine lines through a given point (or subspace con-

tained) in S determines a spread of the strongly regular graph. Clearly, if a geometry

is constructed by taking the cliques deterrnined by subspaces contained in S then the

spreads of the graphs are spreads in the geometry. Moreover, if in this case S can be

partitioned into disjoint subspaces (of the dimension that determines the cliques of G

which are iines of S), then the geometry will be resolvable, Le., it will have its set of lines

partitioned into different spreads.

Among the geometries that we have found that are resolvable, we point out:

CHAPTER 5. ANALYSIS OF SOME OF THE GRAPHS FOUND

The spg(2,1O,l,S); the spread is given by the points of S.

The spg(3,16,1,12); where S can be partitioned into lines in PG(7,2).

The spg(4,25,1 20) ; the spread is given by the points of S.

0 The spg(2,55,1,20); the spread is given by the points of S.

r Mathon's pg(8,20, 2); in this case S can be partitioned into lines in PG(5,3).

m Each of the six geometric structures found in the graphs srg(6256,156,29,42); the

spread is given by the points of S.

5.4 Factorizations of the complete graph into strongly

regular graphs with given parameters

Consider the complete graph Kn2. If g = (n - l ) / t , then it is possible, a t least in principle,

to partition the edges of Kn2 into t graphs of type NL,(n). In this section we mention

sorne of the cases where this is known to be possible; in some other cases it is possible to

decompose Kn2 into graphs of negative Latin square type, not al1 of them with the same

parameters. We end this section by showing how it is possible to factorize K625 into 4

graphs of type NL6 (25 ) .

5.4.1 The graph K256

Consider the cyclotomic scheme C(256,17). The eigenmatrix of this scheme (obtained

by taking the Singer cycle and fusing every 15-th ciass in C(256,l)) is s h o w below.

I t is possible to factorize K256 into 5 graphs of type NL3(16); the first such graph is

obtained by fusing the classes corresponding to columns 1, 6 and 11 in the matrix, and

the others are cyclic shifts of this set of columns. In other words, if we denote the à-th

column of this matrix as q, and the fusion of classes by a + sign, the 5 graphs can be

Sirnilarly, it is possible to factorïze K256 into 3 graphs of type NL5(16), the graphs

k i n g cl + c4 + c7 + cl0 + c13, C* + CS + Cf3 + cl1 + c l 4 and c3 + Q + cg + cl2 + cl5-

5.4.2 The graph Kszs

Consider the cyclotomic scheme C(625,52). The eigenmatrix of this scheme (obtained

by taking the Singer cycle and fusing every 12-th class in C(625,4)) is shown below.

It is possible to factorize Km5 into 6 graphs of type NL4 (25); the graphs can be

espressed as cl + c7, c2 + cg, CQ + Q, cq + ~ 1 0 , CS + cl1 and + c12.

Similarly, it is possible to factorize Ks25 into 3 graphs of type NL8(25); this can

be done by taking the above partition into 6 graphs of type NL4(25) and combining

these graphs in pairs; the following is an example of such a partition: cl + c2 + c~ + c g ,

c3 + cq + cg + and CS + G + + c12. It is also possible to decompose Ks25 into two graphs of type No(25) and one graph

of type XL4 (25): take cl + c 2 + c3 + Q + cll, CS + c7 + c~ + cg + cl* and c4 + clo.

We conclude this section, as promised, with examples of a non-trivial factorization of

into four graphs of type NL6(25).

Consider the eigenmatrix of the cyclotomic scheme C(625,12). This matrix has 52

columns, which can be arranged cyclicly according to the Singer cycle in PG(3,5).

We have found three partitions of the set {O,. . . ,51) (corresponding to the columns of

this matrix) into sets that yield a graph of type NL6(25); therefore each of these partitions

determines a decomposition of Kszs into four graphs of type NL6(25) (or, equivalently,

a partition of PG(3,5) into four two-intersection sets with the same parameters). In the

three cases the partition of the set of columns is cyclic, which implies that the four graphs

in each decomposition are isomorphic to each other. The graphs involved in different

decomposit ions are non-isomorphic to each other: t hey are isomorphic, respect ively, to

graphs G625156,2, G625156,3 and G625i56,6 in our k t .

The actual partitions of the set of rows of the eigenmatrix (and the partition of

PG(3,5) into four two-intersection sets corresponding to graphs of type NL6(25)) are

the following:

Chapter 6

Conclusions

In this chapter we list Our main contributions and explain how to continue this work in

the future.

6.1 Contributions

Our main contribution has been the development of a methodology to search for strongly

regular graphs arising from larger association schemes. -4lthough the ideas of fusing

classes in an association scheme and assuming an automorphism group in order to make a

combinatorial search manageable are not new (for previous work on fusion on association

schemes, see for example [56], [16] and [50]), we have put together a method that allows

u s to try in a systematic way all, or almost all, possible groups of autornorphisms.

Part of our contribution is the result of applying Our methodology to the particular

case of cyclotomic schemes: in this way, we have found many new strongly regular graphs,

corresponding to ten different parameter sets. Here we must also count the discovery of

the geometric configurations with 625 vertices, lines of size 5, 39 lines through each points

and where the number of points of a line collinear with a point outside the line is either

0, 1 or 2, as well as the decompositions of the complete graph K625 into edge-disjoint

copies of graphs of this kind.

6.2 Future work

There are several ways in which the work presented in this thesis can be continued,

estended and improved. We mention some of thein:

1. The fuse algorithm must be irnproved. Such ideas as taking advantage of the

automorphism group of the matrix, and using integer progtamming techniques to

guide the search, must be explored.

2. The methodology should be applied to other association schemes, to search for

strongly regular graphs which are not of cyclotomic type. In particular, this should

be done to find grapbs on numbers of vertices which are not powers of primes.

3. An attempt should be made to find theoretical constructions to express the graphs

found by computer. If possible, such constructions should be generalized to get -

possibly infinite - families of graphs.

Appendix A

Graphs on new parameter sets

This appendix lists the properties of the new graphs that we have found. For each

parameter set (v, k, A, p) , the i-th graph with these parameters is labeled as Gukvi. For

each graph, we show:

the size of its largest clique;

the number No of such cliques through each vertex;

a the size of the translation complement ro of its automorphism group: al1 the graphs

shown here admit the elementary .4belian group on v elements, this group is the

group of translations in PG(n, q); the translation complement is a subgroup of the

automorphism group that together with the group of translations generates the full

automorphism group;

the number of orbits of the translation complement;

the orbit structure of the translation complement.

If the total number N of maximum cliques in the graph is wanted, it can be computed

as N = Nov/s where s is the size of the maximum cliques. If the size of the full

automorphism group is wanted, it can be computed as v Iro 1.

Table A. 1: Graphs with parameters srg(625,156,29,42).

Graph

G625156,1

G625156,2

G625 I 56,3

Max. clique size

5

5

5

No 39

39

39

Orbit structure

4' 12 16~24~48~

1 2 ~ 2 4 ~ ~

12~36' 7z4

lrol 48

24

72

# of orbits

20

28

16

APPENDIX A. GRAPHS ON NEW PARAMETER SETS

Table A.l - continued

# of orbits 1 Orbit s t ~ c ~ ~ ~

Table A.2: Graphs with parameters srg(729,308,127,132).

Graph Max. clique size # of orbits

55

13

62

22

74

64

99

29

44

29

24

58

53

75

51

65

74

50

62

42

38

87

7 1

53

53

67

78

Orbit structure

2448816 1 6 ~ ~ 3 2 ~

1 22242483642964

46824 1 6 ~ ~

1z2 1 6 ~ 2 4 ~ 4 8 ~ ~

2448838 1 6 ~ ~ 4922 1632

210433840 1616

6282 1 2 ~ 16'24~~48~

648412162420

6282 E5 16'24~~48~

1 2 ~ 1 6 ~ 2 4 ~ 4 8 ~ ~

4% l6 1 6 ~ ~

4489 1 6 ~ ~ 264118321626

2343658' 12~~24'~

2'4'8= 1 6 ~ ~

25446131252

23436781 12172419

42 1

42 1 2 ~ ~ 2 4 ~ ~

422036

2544639

254467

23436781 1 2 ~ ~ 2 4 ~ ~

254465 1 2 ~ ~ 2 4 ~ ~

224168 12%

2544621

APPENDIX A. GRAPHS O N NEW PARAMETER SETS

Graph

Table A.2 - continueci

Max. ciique sizç --

# of orbite

APPENDIX A. GRAPHS ON NEW PARAMETER SETS

Table A.2 - continued

Max. clique siz # of orbit

APPENDIX A. GRAPHS ON NEW PARAMETER SETS

Table A.2 - continueci

Max. clique size

9

9

9

1 O

9

9

9

9

9

9

9

9

9

9

11

1 O

9

10

9

9

9

# of orbits Orbit structure

422036

Table A.3: Graphs with parameters srg(729,336,153,156).

Max. clique size # of orbits Orbit structure

APPENDIX A. GRAPHS ON NEW PARAMETER SETS

Table A.3 - continued

Max. clique size # of orbits

APPENDIX A. GRAPHS ON NEW PARAMETER SETS

Table A.3 - continueci

Max. clique size - -

Orbit structure

24488n 1 6 ~ ~ 28418848 1616

25446131252

23436781 12172419

APPENDIX A. GRAPHS ON NEW PARAMETER SETS

Table A.3 - continued

# of orbits Graph

G729336,71

G729336,f2

G729336,73

G729336,74

G729336,75

6729336.76

G729336,??

G729336,78

Table -4.4: Graphs with parameters srg(1024,231,38,56).

Max. clique size

12

11

11

11

10

12

11

11

# of orbits

20

18

14

8

1 Orbit structure 1

Table A.5: Graphs with parameters srg(1024,297,76,90).

Table A.6: Graphs with parameters srg(1024,330,98,110).

Graph

G1024297,1

G1024297,2

Graph

Gl024330,l

G1024330,2

G1024330,3

G1024330.4

G1024330.5

G1024330.6

G1024330.7

G 1024330~8

G1024330.9

G1024330.10

Max. clique sue

16

16

16

16

16

16

16

16

16

16

Max. clique sue

8

8

Orbit structure

3'12' 1 ~ ~ 2 4 ~ 3 0 ' 6 0 ~ 1206

15'21'42' 105'2104

3'6'g4 1 2 ~ 18173618

3'6'g4 1 2 ~ 18173618

3 ' 1 5 ~ 3 0 ~ ~

3' 1 5 ~ 3 0 ~ '

3'

3115630760'2

3l 15fj8

352148

No 336

315

Iro] 18

18

# of orbits

65

76

Orbit structure

336496 1 8 ~ ~

3762924 1843

APPENDIX A. GRAPHS ON NEW PARAMETER SETS

Table A.6 - cont inued - - --

Max. clique size

16

16

16

16

16

16

16

16

8

16

16

16

16

# of orbits Orbit structure

3L6221'242L8

3112'212425849

3'12'212425849

3'6221'24218

3'6221'24218

3' 1 2 ' 2 1 ~ 4 2 ~ 8 4 ~

3' 1 5 ~ 3 0 ~ '

3' 1 5 ~ 3 0 ~ ~

3' 1 5 ~ 3 0 ~ '

3' 1 5 ~ 3 0 ~ '

31156303L

3 '15~30~ '

3' lS63o3'

Table A.7: Graphs with parameters srg(1024,363,122,132).

Graph - -

Max. clique size

16

16

16

16

16

16

16

16

16

16

16

16

16

8

16

16

16

# of orbits Orbit structure

336'3 1 2 ~ ~ 2 4 ~ "

3162 1 2 ~ ~ 2 4 ~ ~ 4 8 ~

3L659101850

3' 1 5 ~ 3 0 ~ '

3762924 1 8 ~ ~

3165920 1 8 ~ ~

3' 1568

3' 1563019606

3' 15303019

3' 15fj8

3 l ls2O 3 0 ~ ~

3 '15~30~ '

3' 1 5 ~ 3 0 ~ '

3 l 1 5 ~ 3 0 ~ '

3 '15~30~ '

3' 1 5 ~ 3 0 ~ '

3' 1 5 ~ 3 0 ~ ~

Table A.? - continueci

Graph

G1024363,18

G1024363,19

G1024363,20

G1024363,21

G1024363,~

G1024363.23

Max. clique size

16

16

16

16

16

16

# of orbits Orbit structure

3' 15~30~'

3'15~30~'

3l15~30~'

3'15~30~'

3' lS63o3'

3' 15~30~'

Table A.8: Graphs with parameters srg(1024,396,148,156).

Graph

G1024396,l

G 1024396.2

G1024396.3

G1024396,4

G1024396,5

G1024396,6

G1024396.7

G1024396,8

G1024396,9

Max. clique size

16

16

16

16

16

16

16

16

16

Iro.ol 1 # of orbits

Table A.9: Graphs with parameters srg(1024,429,176,182).

APPENDIX A. GRAPHS ON NEW PARAMETER SETS

Table A.9 - continued

# of orbits

41

41

38

38

38

38

38

Table A.10: Graphs with parameters s+g(1024,462,206,210).

Max. clique size

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

APPENDIX A. GRAPHS ON NEW PARAMETER SETS

Table A.10 - continueci .. .. .-

Max. clique size

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

--

Orbit structure

APPENDIX A. GRAPHS ON NEW PARAMETER SETS

Table A. 10 - continueci

Max. ciique size # of orbits

APPENDIX A. GRAPHS O N NEW PARAMETER SETS

Table A-IO - continuai

Graph

G102&62,84

lrol 42

36

36

36

36

36

36

36

- - -

Max. clique size

16

No 259

# of orbits

33

38

38

38

38

38

38

38

Orbit structure

316221'24218

3'6'g2 12218103622

3'6'g2 12218103622

316'g2 122181036n

3'6192122 181°36=

3'6'g2 1 2 ~ 1 8 ~ ~ 3 6 ~ ~

31619212218103622

3'6'g2 1z2 1 8 ~ ~ 3 6 ~ ~

Appendix B

The new graphs

This appendix gives a concise description of sorne of the graphs listed in the previous

appendk Each of the graphs listed here is represented by a matrix; each column of

this matrix shows the hornogeneous coordinates of one of the points that make up the

two-intersection set in the corresponding PG(n - 1, q). Alternatively, we can think of

the rows of this matrix as representing a codeword in IF: (where N is the size of the two-

intersection set); the full matrix represents a basis of the two-weight code determined in

the way described in [16, Section 21. The graphs on 625 and 729 vertices are presented as

vectors over IF5 and IF3 respectively. The graphs on 1024 vertices are presented as vectors

over IF4, considered as formed by O, 1, and two elements a and b such that a* = b = a+ 1.

-4s was said above, only sorne of the graphs are shown; this is due to the large number

of these graphs, that would make a complete listing span over scores of pages. In par-

ticular, we show only 10 graphs for each of the following classes: srg(729,308,127,132),

srg(729,336,153,156), srg(1024,330,98,110), srg(1024,363,122,132), srg(1024,429,176,182)

and s~g(1024,462,206,210); for the other four parameter sets we show al1 the graphs

(which are less than 10 in each of these cases).

.A complete Iist of the graphs is available online a t

<~~~:http://ww.cs.toronto.edu/-ldissett/new~aphs~~

Graphs with parameters srg(625,156,29,42):

APPENDIX B. THE NEW GRAPHS

Graphs with parameters srg(729,308,127,132):

ooooooi i i r 1 i i 1 1ioooooooo00ooo~ 11 11 111 11222222200000000000001 i 111 1 i 11 122222222 00000000000000001111111111111111111111111111110000000000000~00000000000~0000 00000000000000000000000000000000000OOO00000001111111111111111111111111111111

APPENDIX B. THE NEW GRAPHS

APPENDIX B. THE NEW CRAPHS

APPENDIX B. THE NEW GRAPHS

Graphs with parameters srg(729,336,153,156):

APPENDIX B. THE NEW GRAPHS 98

Graphs with parameters srg(1024,231,38,56):

APPENDIX B. THE NEW GRAPHS

Graphs with parameters srg(1024,297,76,90):

OblbllablbOOlOaOlblaabablOllbObOlaabbO1abOablbO1a 001lb00a001bb001aabbOOllabbOOllabbOOlaab11a bbbbb000111llaaaaaaabbbbbbbOOOOOOOlllllllaaaaabbb lllllaaaaaaaaaaaaaaaaaaaaaabbbbbbbbbbbbbbbbbbbbbb 1111111111111111111111111111111111111111111111111

bbabbbbOlObalObOalabOa10bObOa010bOaaObOaOlbbOlObO blaablabb01abO1bbll1aabûOlabb11aaOOab011aaaOllaab laaaabbbbOOOOllllaaaaaabbbbbb0000111laaaaaabbbbbb llillllllaaaaaaaaaaaaaaaaaaaabbbbbbbbbbbbbbbbbbbb 1111111111111111111111111111111111111111111111111

Graphs with parameters srg(1024,330,98,110):

1 11 1 1 11 11 1 laaaaaaaaaaaaaaaaaaaaaabbbbbbbbbbbbbbbbbbbbbb 1111111111111111111111111111111111111111111111111111111

APPENDIX B. THE NEW GRAPHS

Graphs with parameters srg(1024,363,122,132):

APPENDIX B. THE NEW GRAPHS

blbOaOlb1 bOlbbaaOalaûûlabObababaObbO1ababab1lab1a1Obûb1O lbla O l b 0 0 a a b 0 0 1 1 1 a b 0 1 a b b O l 1 1 1 a a b b 0 1 a b b O 1 l l aaabbbbbOOOOOOOlllllaa~a~~bbbbbOOO0000001111111aa~abbbbb llllllllaaaaaaaaaaaaaaaaaaaaaaaaaabbbbbbbbbbbbbbbbbbbbbbbbbb 11111111111111111111111111111111111111111111111111~111111111

Graphs with parameters srg(1024,396,148,156):

APPENDIX B. THE NEW GRAPHS

APPENDIX B. THE NEW GRAPHS

Graphs with parameters srg(1024,429,176,182) :

APPENDIX B. THE NEW GRAPHS

Graphs with parameters srg(1024,462,206,210):

APPENDIX B. THE NEW GRAPHS

blaOblbOaOlObablbOlablM)bO11aab01abOaablaO~bOa1bablbabOlab1aablblaO1abOaObOl OllbbOOllbbOOaabbOOOOllaabbOO1laaaabbOOllaabbOO11~OO11a~abbOObbOOaaaallaabb 11111aaaaaabbbbbb00000000001111111111aaaaaaaabbbbbb00000000001111aaaaaabbbbbb l l l l l l l l l l l l l l l i laaaaaaaaaaaaaaaaaaaaaaaaabbbbbbbbbbbbbbbbbbbbbbbbbb 11111111111111111111111111111111111111111111111111111111111111111111111111111

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