Combinatorial Mathematics: Proceedings of the Second Australian Conference

157 Pages·1974·1.998 MB·English

by  D. G. Higman (auth.), Derek A. Holton (eds.)

Lecture Notes ni
Mathematics
Edited by .A Dold and .B Eckmann
403
Combinatorial Mathematics
Proceedings of eht Second Australian Conference
Edited yb .D .A Holton
Springer-Verlag
Berlin (cid:12)9 Heidelberg (cid:12)9 New York 1974
Derek A. Holton
University of Melbourne
Dept. of Mathematics
Parkville, Victoria
Australia
Library of Congress Cataloging in Publication Data
Australian Conference on Combinatorial Mathematics, 2d,
University of Melbourne, 1973.
Combinatorial mathematics; proceedings of the
second Australian conference.
(Lectures notes in mathematics, 403)
i. Combinatorial amalysis--Congres se .s .I Holton,
Derek A., 1941- ed. II. Title. III. Series:
Lecture notes in mathematics (Berlin) 403.
QA3.L28 no. 403 QAI64 510',8s 511'.6
74-14845
AMS Subject Classifications (1970) 05Bxx, 05Cxx, 20B25, 50Dxx
ISBN 3-540-06903-8 Springer-Verlag Berlin (cid:12)9 Heidelberg (cid:12)9 New York
ISBN 0-387-06903-8 Springer-Verlag New York (cid:12)9 Heidelberg (cid:12)9 Berlin
This work is subject to copyright. All rights are reserved, whether the whole
or part of the material is concerned, specifically those of translation,
reprinting, re-use of illustrations, broadcasting, reproduction by photo-
copying machine or similar means, and storage in data banks.
Under w 54 of the German Copyright Law where copies are made for other
than private use, a fee is payable to the publisher, the amount of the fee to
be determined by agreement with the publisher.
(cid:14)9 by Springer-Verlag Berlin (cid:12)9 Heidelberg 1974. Printed in Germany.
Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
FOREWORD
These are the proceedings of the Second Australian Conference on
Combinatorial Mathematics. It follows the first such conference held in Newcastle,
New South Wales in 1972, the proceedings of which were published by the
University of Newcastle Research Associates Limited (TUNRA) and can be obtained
from them.
I would like to express my thanks to Douglas .D Grant, M. Adena, and
K.McAvaney for their help in organising the conference and to J.J. Cross for his
assistance with accommodation in Queen's College.
The departmental secretaries Shirley Flinn~ Irene Dickson and
Janine Malley also deserve special mention for their work in preparing for the
conference and their typing of manuscripts.
August, 1973 Derek A. Holton
TABLE OF CONTENTS
Invited Address
D.G. HIGMAN
Coherent Configurations and Generalised Polygons . . . . . . . . . . . .
Contributed Papers
M.A. ADZ~A, D.A. HOLTON and P.A. KELLY
Some Thoughts on the No-Three-in-Line Problem . . . . . . . . . . . . .
K.K.H. BUTLER
A Moore-Penrose inverse for Boolean Relation Matrices . . . . . . . . . 18
D.D. GRANT
The Stability Index of Graphs . . . . . . . . . . . . . . . . . . . . . 29
D.A. HOLTON
Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
L. JANOS
An A~lication of Combinatorial Techniques to a Topological Problem . . 56
O.H. KEGEL and A. SCHLEIRMACH~R
Embeddingsof Projective Planes . . . . . . . . . . . . . . . . . . . . 16
C. LITTLE
Extensions of Kasteleyn's Method of Enumerating the 1-Factors of
Planar Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
P.J. LORIMER
Class of Block Designs Havin~ the Same Parameters as the Design
of Points and Lines in a Projective 3-Space . . . . . . . . . . . . . . 73
K.L. XcAVANEY
Counting Stable Trees . . . . . . . . . . . . . . . . . . . . . . . . . 79
I.A. PECKHAM
ffhe Ha~iltonian Product of Graphs . . . . . . . . . . . . . . . . . . . 86
A. RAHILLY
Derivable Chains Containing Generalised Hall Planes ........... 96
L. ROBERTS
Characterisation of a Pregeometry by its Flats . . . . . . . . . . . . 101
A. PENFOLD STREET
Eulerian Washing Machines . . . . . . . . . . . . . . . . . . . . . . . 105
A. PENFOLD STREET and E.G. WHITEHEAD Jr.
Sum-free Sets, Difference Sets and Cyclotomy . . . . . . . . . . . . . . 109
G. SZEKERES
Polyhedral Decomposition of Trivalent Graphs . . . . . . . . . . . . . . 125
D.E. TAYLOR
Graphs and Block Designs Associated with the Three-Dimensional
Unitary Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
J.S. WALLIS
Williamson Matrices of Even Order . . . . . . . . . . . . . . . . . . . 132
W.D. WALLIS
Supersquares . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
A. WERNER and R.J. BAXTER
A Combinatorial Measure of Structure for Models of Physical
Phenomena . . . . . . . . . . . . . . . . . . . . . . . (paper not included)
LIST OF PARTICIPANTS
M.A. Adena, Melbourne University, Vic.
S.J. Anderson, W.R.E., Salisbury, S.A.
.A Brace, Canberr& C.A.E., A.C.T.
K.K-H. Butler, Pembroke State University, U.S.A.
.P Cain, University of Newcastle, N.S.W.
J.J. Cross, University of Melbourne, Vic.
.E Cousins, University of Newcastle, N.S.W.
.H Enomoto, University of Tokyo, Japan.
D.D. Grant, University of Melbourne, Vic.
J.R.J. Groves, University of Melbourne, Vie.
.S Groves, Australian National University, A,C,T.
V.W.D. Hale, University of York, England.
D.G. Higman, University of Michigsa, U.S.A.
D.A. Holton, University of Melbourne, Vic.
.L Janos, University of Newcastle, N.S.W.
J.N. Kapur, Meerut University, India.
O.H. Kegel, Queen Mary College, London, England.
.T Klemm, Gordon Institute of Technology, Geelong, Vic.
.C Little, Royal Melbourne Institute of Technology, Vic.
P.J. Lorimer, University of Auckland, New Zealand.
.W Magnus, New York University, U.S.A.
K.L. MeAvaney, Gordon Institute of Technology, Geelong, V~
B.H. Neumann, Australian National University, A.C.T.
A.G. Pakes~ Monash University, Vic.
.I Peckham, Royal Melbourne Institute of Technology, Vie.
.A Rahilly, Sydney University, N.S.W.
.L Roberts, University of Tasmania, Tas.
.D Row, University of Tasmania, Tas.
G.W. Southern~ University of Newcastle~ N.S.W.
.A Street, University of Queensland, Q'Id.
.G Szekeres, University of New Sou~h Wales, N.S.W
D.E. Taylor, La Trobe I.~'~T J~S_~y,~x vis.
.J Wailis, Australian National University, A.C.T.
W.D. Wallis, University of Newcastle, N.S.W.
.A Werner, W.R.E., Salisbury, S.A.
M.J. Wicks, University of Singapore.
.K Yamaki, University of Osaka, Japan
COHERENT CONFIGURATIONS AND GENERALIZED POLYGONS
D.G. Higman*
i. Coherent configurations.
This concept abstracts certain aspects of the combinatorial structure in-
duced in a set by a group acting on it. Precisely, a coherent configuration
(X,~) consists of a finite nonempty set X and a set ~of binary relations
on X satisfying the following four conditions.
2
)I( ~ is a partition of X
)II( I = {(x,x) I x~_ X} is a union of members of ~.
(III) For f~ ~, f~ = {(y,x) I (x,y) ~ f} ~ .
)VI( For ,f g, hE ~ and (x,y) ~ h, the number
f(x) N g(Y) I is independent of the choice of (x,y) ~ h .
Here f(x) = {y~ X I (x,y)~ f} = the set of vertices adjacent to x
in the graph (X,f) . The number r = 181 is called the rank, and the
numbers afgh = I f(x) ~ gg(Y) I , ,f g, h~ ~and (x,y)~ h , are the
intersection numbers.
If a group G acts on a set X and ~ is the totality of G-orbits in
X 2 (cid:12)9 then (X,~) is coherent. We refer to this situation as the ~ case.
A homogeneous configuration is a coherent configuration such that I ~ .
In the gronp case, homogeneity is equivalent to transitivity. The homogeneous
configurations with all f~L~ symmetric are equivalent to the association
schemes of Bose and Mesner. It turns out that a coherent configuration can
be regarded as a collection of homogeneous configurations linked together in
a "coherent" fashion. Projective designs, partial geometries and familes of
*Research supported in part by the National Science Foundation.
linked projective designs are equivalent to certain imhomogeneous coherent
configurations.
The results of a systematic study of coherent configurations will appear
in a series of papers 2. Here we will state three basic results (i), )2( and
)3( from Part I of the series and describe an application )4( from Part IV.
To do this we need to define the ad~acenc algebra of a coherent configuration.
In the group case this is the centralizer algebra of the permutation repre-
sentation. On the one hand, coherent configurations provide a combinatorial
setting for centralizer ring theory of permutation representations, and on
the other, we are able to apply methods of centralizer ring theory to study
coherent configurations.
It follows from axioms )1( and (IV) that the totality C of matrices
: X 2 § (cid:12)9 such that the restriction of ~ to f is constant for all
f ~ i s a subalgebra of the "algebra of all matrices with coefficients in the
complex number field ~ having rows and columns indexed by X . Namely,
C has basis ~ = {~f ; f~ ~} , where ~f = the characteristic function
of f ~ X 2 = adjacency matrix of the graph (X,f), and the structure constants
for C with respect to this basis are the intersection numbers. We call
C the ~djacency algebra.
By axiom (II), C contains the identity matrix, and since C is closed
under the conjugate transpose map ~ ~ ~ by axiom (III), it is semisimple.
Let A. , 1 < i < m , be the inequivalent absolutely irreducible repre-
l
sentations of C , and let ~i be the character afforded by a. and e.
l l
its degree. The standard character ~ is defined by ~(~) = trace ~ , ~ ~ C ,
and we have ~ = im~ = 1 zi ~i with z i a positive integer, 1 < i < m
We call the z i the multiplicities.
The Schur relations involve the coefficient functions of the A. . Let
l
us write A )~( = (a~j(~)) for ~ C and list the aij : a2,..
al , (cid:12)9 a r (cid:12)9
(The number of these functions is ~ i = 1 e 2 i , which is equal to the rank r.)
If al = aij , put a = a.. and h~ = z For f~ ~, put ~f =
3~ ) fv"
The Schur relations can now be written in the form
)i( Z a )f~#( = 6 i
f~--~ a~ (~f) ~ h%
They imply the orthogonalit relations
E ~ e
)2(
f~@ ~(~f)~(~f) =~ z
The irreducible representations and characters are determined by the
intersection numbers. The multiplicities can be computed from the characters
by )2(
Because of its relation to the similarly named result of L.L. Scott, Jr.
4, we refer to the following result )3( as the Krein condition for config-
urations. We assume that A (~*) = a (~)* for 1 < ~ < m and ~ C ;
this can be arranged by effecting a complete reduction of C by a unitary
matrix.
)3( Choose ~ and ~ , 1 < ~, V < r , such that ~ =
and ~ = ~ . If ax = (cid:127)j~a put
#
c~ Z al(~f) a (~f) a (~f)
f~ ~ JfI 2
(cid:12)9 ~
Then for 1 < d < m , C = (c..) is a positive semidefinite hermitian matrix.
l