UNIVERSITY OF NORTH CAROLINA
Department of Statistics
Chapel Hill, N. C.
STRONGLY REGULAR GRAPHS, PARTIAL GEOMETRIES AND
PARTIALLY BALANCED DESIGNS
by
R. C. Bose
March, 1963
Grant No. AF AFOSR-60-2l
This paper introduced the concept of a partial geometry, which
serves to unify and generalize certain theorems on embedding
of nets, and uniqueness of association schemes of partially
balanced designs, by Bruck, Connor, Shrikhande and others.
Certain lemmas and theorems are direct generalizations of
those proved by Bruck ;-57, for the case of nets, which are
a special class of partial geometries.
This research was supported in pal~ by the Mathematics Division of the Air
Force Office of Scientific Research.
Institute of Statistics
Mimeo Series No. 358
•
strongly Regular Graphs, Partial Geometries
and Partially Balanoed DeSigns1)
by
n.c.
Bose
2)
University of North Carolina and University of Geneva.
O. Summary.
•
This paper introduo8l'l1 the oonoept of a Esrtial aeometr;x:,
whioh serves to unify and generalize oertain theorems on ombedding
of nata, and uniqueness of association schemes of partially balanced
deaigns, by Bruck, Connor, Shripkhande and othere. Certain lemmas
and theorems are direot generalizations of those proved by Bruck [5],
for the case of nets, which are a speoial olass of partial geometries.
1) This rescaroh wall supported in part by the Uni ted St[\tel~
Air Foroe under grant no AF Alt'OSR-60-21, moniterod by the Office of
Scientifio Research.
The author also wishes to aoknowledge, that some of the
ideas on whioh the present paper is based originated during dlaousaions at the Symposium on Combine torial Jiluthemstio8 held at the
Rand CorporationJSanta Monica,California,in the smarnor of 1961, in
whioh the author participated~
2) Now visiting profossor to the University of Geneva, on
leave from the Univera1ty of North Carolina,
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•
1. Introduotio!!._
We use graph theoretio methods for the study of
aa~ooiation
sohemes of partially balanoed incomplete blook (PBIB) designs_ For
this purpose it is convenient to switch from graph theoretic language
to tho language of designs and vice versa as neceouary.
AD we ahall be ooncerned with finite gra.phs only, we shall
use the word graph in the senae of finite graphs_
to
n
1
A {;raph G is said to be re&'Ular if ~ach vertex is joined
other vertices, and unjoinod to n
other vertioes. Clearly
2
1:t:.nJ
If further any j(dned verticos of
A
•
G, are both joined to
exaotly P~1 other vertices, and any tv/o unjoined vertices are both
2
.
joined to exactly P
other vertices, then the graph G 1s defined
11
to be stron;;ly:.regular, with parameters
The concept of stronGly regular eraphs i6 isomorphic with
the concept of association schemes of PDID doolgns (With two asnociand Shiw\.l!1oto [4]Such a scheme they definedaa a scheme of rolations between v treatemsnts such that (1) any two objects are eithel' first alHH)ciates
ate olasses), whioh was first introduced by
JJOI3€l
or saoond associatos (ii) each treatement haa n
1-th associates
1
(i • 1,2) (ii1) If two treatments are l-th associates, then the
nUlUber of treatments OOlllliJOn to tho j-th assooiates of the first and
k-th associates of tbe second 10
IN
P~k Bnd ia indopendont of the
1
1
pair of treatmonts with ?hiCh we start. Alao Pjk· Pkj Dose and Clutworthy [1] showod that it is unneOO:.Hlnry to
assume the oc.ue tan~y of all the p i k 'a _ If we aUBulIIctha t n , n ,
j
2
1
1
2
1
1
P11 and P11 a.re oonsta.nt, then the oonstancy of the P12' P21 ,
1222
1
1
2
2
P22' P12' P21 , P22 follows and P12· P21' P12 • P21 •
It we now identify the v treatments of tho association
soheme with the v vertices of a graph 0, and oonoidcr two vortices
-,-
•
aa joined or unjo1ned aooording as the corresponding treatments are
first 01' seoond associa.teo, it is olear that a. strongly re{;,"Ular
graph G with the parameters (1.2) is isomorphio with an a.ssooiation
soheme with the same parameters.
We ha.ve introduoed for the first t1nle in this paper the
oonoept of a
l?~rtial
i::eomotr;z:.
A pa.rtial geomotry (r,k, t) is a sys'~efll of undefined points
and
~ines,
"1 -
A4.
and an undefined relation incidenoe satisfying the axioms
To avoid oumbersome ex.pl'esaion we may use standard geometric
language, 'llhuB a point incident with a line may be said to lie on it
and tho line may be said to pass through the point. If two linea are
inoident with the sallie point, we aay that they intersect.
A1. Any two po1ntH ar,a'incident with not more than one line.
A2. Each point is incident with r lines.
A3. Eaoh ~ine is inoident with k points.
A4. It the point P is not inoident with the line 1, there
paas through P exactly t
lines (t ~ 1) intersecting 1We ahow that th\.' number of points v
and the number of
lines t in the partial goometry (r,k,t) are given by
v • k [(r - 1)(k - 1) +
t. l' [(1' - 1)(k - 1) +
t]/
t]/
t,
t.
G ot a partial geometry in defined as a graph
The graph
whose vertices correspond to the points of the geometry, and in
which two vertioes are joined or unjoined nccording as the corresponding points are incident or nOll-illc!clent with a OOIilmon line.
We then provo I
Theorem. 'l'he
{~raph
G of
(l.
partial georllstry
(r,k, t)
io
8 tront;;ly rogular with paraHle taro
n
P
whure
l~t~r,
t
11
•
(k - 1),
n 2 • (1' - 1)(k - 1)(k - t) / t,
• (t - 1)(r - 1) + k - 2,
P~ 1 • rt.
l'
1~t~k.
Using certain theorems of 11080 and Meanor [~] relnting to
- 4 -
•
association achemoa, we derivo
Theorem. A nucessary condition for the existence of a partial geometry (r,k,t) ia that tho number
,
is integral.
A strongly reL,'ular graph with parameters (1.t), (1. t) and
for which (1.7) is ao.tisfied will be defined to be a pl3eudo .. ,~~!netr;~
gra.ph with oharaoter-iatico (r,k,t). Such a gra.ph way or ma.}' not be
the graph of a l)u,rtial ~eometry (r,k,t). It ia therefore of interest
tc stuclythe oonditions undor whioh a atrongly reg1J.lnr graph, and. in
partioular a. pf:H.luclo-(;·eolUatric gra-lih wi th chHI'e.oteriBtics (r, k, t) ia
the graph of a partia1 (;oometry (r,k,t).
A subset of
vertic~6
joined is called a .2J.l.que of
G. Whon
geometry (r,k,t) ttlHJro will exis'& in
K , K , ••• , K])
2
1
a,
of a graph
is tho g'rapll of u. partial
G
G
any two of whioh are
a net ~ of distinot oliques,
corresponding to the lines of the
L:Qom~try
satis-
fying the tollowing Bxioms
A*1. Any two joinod vertioes of
G
are contained in one
and only one oli:lue of ~ •
A*2. ':;;/3,ch vertex of
A*:5. gach cliq,ue of
A*4. If
Xi
to
P
G
L
is can tained in
contains
ia a vertex of
of ~ there are cxne tly
p(!. 1, 2, ••• , t).
Henoe anj' Graph
t
G
k
cliques of ~.
r
vertices of
G.
G not contained in a clique
vertices in
K!, which are joined
in which tlwre axis ta
Ii
set
2. of oliques
K1 , 1(2' ••• , Kl/,satisfying axioms A*1 to A*4. is the graph of s.
pr.rtial geometry (1', k, t). In fact G together with tho 01 iques of L
is 1aomorp/l1e to eo ps.rtial gemiJO-tz'y (I', k, t) the' vertic os of G
correspon1tins' to the points nnei the cliques of ~ cox-responding to
the linea of the geometry. Such a graph will be called
.etlomo~ri8nble
(r,k,t).
One may oonsider lsrapha in whioh there Qxists a. eet
cliques
1\1' K , ••• , K])
2
of
eu Liofying one or luore but not all of the
- 5 -
•
A" A2, A'3' A4.
axioms
Thus a previouD rcoult duo to Bosc and
ClatwortllY [1], 1s equivalont to tho following I
~h()orom. If in n strong1y regula:!' gra,ph
Z
a set
of
K , K , ••• , K , sn tisfyinf<' tilO D.xioms A" A~,
2
b
1
k ~r, 'then tho g'l'aph ia geom~!~risablo (r,k, t), the verti-
of oli(luea
A;. aud if
008
0, thero exists
G, and tho oliquEHJ of
I
being tho points and linea of the
oorrespond.ing g-eometry.
We further prove
Theorem. If in a pa(;udo~geometrio gral1h wi th ohl.1raoteristics
(r,k.t), there exists a set ~ ofoliquea K , K , ~ • ., K , satisfying
2
h
1
axiomB At and A~, and if k > r, then G ia geometrisable (r,k,t),
the vertioos of G, and tho oliques of L being the points und lines
of the correslJonding guometry.
~here
ara many intereotine examples of partial geometries
some of which aro given in scotian 7. In partioular the IJH.rtial
geometry (r, k, t) becomes a not
at:
t • r-lo A pSfJudo-geoLwtrio graph
d.egreo
r
Hod order
wi tIl cha.rfl-oteri£d,ics
be defined to bo a JJso}Hlo-rw·t graph, of dOf;reo
liruck
[5J
k
when
(;r, k, r-1) may
und ordor
k.
haa proved a serios of Lelnraan for poeudo-not graphs and,
1n pa.rticular has ahovm that
order
r
k
0.
pseudo-net gra.ph of degree
rand
1s geomotrisable (r,k,r-1) if
The 13p00ial anoe
r . 2, was proved by Sh:dkh£l.ndo [19J. In this
paper we give the following
Theorem. A
gonerali~ation
pB~udo-geomotric
(r,k,t) ia geomotrieable
(r,k,i)
of Bruck's result.
graph with characteristios
if
k.>t[r (r-1) + t (r+1)(r 2_ 2r + 2)J.
We have proved a oarien of Lemwaa whioh ara tiireot ge:nel'alizationa of the Le!:iItlas u9t:d by Druok for hio pl'oof. In faot it is
uurprbing how smoothly the teohnitLue devised by Eruck for the apeoial cnew of natu t worka in tho ganeral oaso.
- 6 -
•
III particular the oonoept ot grand cliques introduced by
Bruck for the oase of nots is oapable of easy generalization. It
G
18 a pfJeudo-g~oul"1tric era.ph with oharaoteristio6 (r,k,t), then a
major o.119ue of
G is defined us a olique whioh con talns at le,-st
k - (r_1)2 (t-1) pointe. A major clique whioh 1s aleo maximal is
definod as a grand clique.
Given a pBeudo-g~ometrio graph G with characteristics
(r, k, t) the set of grand oliques in unambiguously defined. 'So may
take this set tu be the 80t L and. enquire under wile t circulllstances,
the axioms A*l - A*4 will be satisfied. We tben ahow that (1.10)
is a suffioient cOndition fOl' this.
A ps~udo-guometr1c graph with characturiotioa (r,k,t) has
the sarna par'a.i,latera as tho triangular assooiation schame if we take
r'. t • 2, k. n-1. Substituting thes$ values in (1,10) we gat
n > 8. Thus i'or these spaoial values of r,k, tour rOBul t is cq,uivalent
to the uniq,uenesB of the tria.ngular Bollome for n > 8, a reaul t tiI'st
proved by Gonnor [9]. In faot our result may be in~erproted ae a
genoralized uniqueness thooram as explained in Btiotion 12.
A net of degree r and order k is defined to have defioiency d. r + 1 - k. Bruok [5J ahowed that a net of order k and
defioiency d oan be oompleted to an affine plane by tho addition
01' new lines, if
We genoralize Bruck's reault to the following I
Theorem. Given a partially balanoed inoomplete blook (pnID)
design (r, k, A1 , A2 ),
parameters
.\1> A , based on an assooia.tion soheme with
2
n 1 • (d-l)(k-1)(k-t) / t,
n 2 • d (k-1),
P~1 • [(d-l)(k-1)(k-t) - d (k-t-1) -
t] I
t,
P~1 • (d-1)(k-t)(k.t-1) / t,
we oan vxtend the
defl1~11
by adding new blooks, containing the same
- 7 -
•
treatmen tu, in Guch
Il
way tha.t the Qxtended design is a br.danced
inooID}leto block (DIB) de8i~I, with r O • r + d (A - A2 ) repli1
oations, block size k and in whioh'~very pair of treatments occur
together in
A blooks.
1
2. Stron«ly
grapho.
A finite graph a conei·sts of a tini to set of v vertice8.
and a rii.11tion adjacimol such that any two distinct vorticos of a
may be either adjacent or non-adjaoent. Adjaoent vertices may be said
re~lar
to be joineg and non-adjacent vertioes to be unJoine4. We shall only
be ooncernedwith finite graphs only. and use the word graph in the
senBO of finite graphs.
The graph G i8 said to be regular (of degree n ' it eaoh
1
vertex of G is Joined to exactly n
other vertices. In this ca.e
1
eaoh vertex will be unjolned to exaotly n 2 other vertioeo. where
A reb~lar graph G will be 8ald to be strongly regular
it (1) any two vertices .hich are Joined in G, are both 8imultaneously joined to exactly P~1 other vertices (11) any two pairs ot
vertices whioh are'unJo1ned in 0. are both simUltaneously joined
to exactly
P~1 vertices.
A 8 trongly regular graph
1
G thus depends on tour
pal'Bllle-
2
n 1• n 2 , P11 and P11' the number ot vertioes being given by (2.1).
Let two vertices of a strongly rogular ~raph G be oalled
tir~t aaoociate8 1f the¥ are Joined, and seoond associates if they
are unjo1ned. If the vertico8 Q and 4> of 0 are i-th ap'8001,,,toB,
tera
.e aha.ll denote by P~k (Q, cI» the number of vertioe8 whioh ara j-th
aS80ciates of g and k-th assoaiates of
From definition the
+.
nwaber P~1(Q.+) 1s 1ndopond~nt of the pair
are i-th associates. Thus
Q.4>
80 long as tlley
i .. 1. 2.
- 8 -
•
We allall ehow that a similar a1 tua tiou prevrdla with res-
P~k (e,
pect to all the numbers
+),
eo that we oan Vlri te
and that
This tallows from the following theorems proved by fioue
and Clatworthy
1, in conneotiol1 with partiall:l balanced inoomplete
block (f'IiIB) dosigns (it VlO identify treatmcntu with verticofl).
'I'ht:lOrelll 2.1. Let thoro exiet a relationstLip of aU8ociation
between every pair
~nong
v
treatmonts satisfying the oonditions
I
(a) Any two troutmants Brc either tirst a6socintas or
second B~eooiates (b) Eaoh troatment has
first and n
seoond
1
2
associates (c) Por allY pair of treatments which ure first a~HJociates
1
the numbor P11
of troatments common to the first uGBooiateo of the
n
first snd the first associates of the second 10 inJependent of the
pair of treatments with which we start.
Than, for every pair of first aasociates muong the v treat1
1
1
1
1
meats tho numbers P12' P21 and P22 al'e aOrlS tall ts, and p12 • P21.
Theorem 2.2. Let there oxist a rl;.:latiolHjldp of ansooia tion
between every pair among
Ca)
AnJ
two
V
treatments satisfying tbe oonditions
I
treatments ure either first asoocintes (b) Each
treatmont hao n
first assooiates and 112 second associatos
1
(0) }t'or auy p:::ir of treatments wJdch al'e seoond a.iHocintes, the l".umber
P~ 1
of treHLHOt1ta OOllunon to the first aosociates of the firat and
tho firat
aB~ociutes
of the seoond la independent or the
~air
of
treatments with whioh we start.
1111(~n,
fo.L'
pail' of second l:w6ociataa cU;lon~ the v
222
P12' P21 anti P22 1:1.1'0 conntcmts, and
ev(~r:t'
traa. tuam ta the numbers
2
2
P12 - 1)21·
It al)poars from the proof of Bose llnd Clutworthy
tllftt
- 9 -
•
(2.~)
1
1
1
P12 .. n 1 - P11 - 1 .. P21 ,
(2.4)
P12 • n 1 - P 11 • P21'
2
Actually the relations
lier by Bose and Nair
[3].
(2.3). (2.4)
were obtained much ear-
but 1n their formulation they started w1th
the constancy of all the numbers
e8IJY to prove as
2
2
P22 .. n2 - n 1 + Pl1 - 1.
2
2
1
1
P22 • n2 - n 1 + Pl1 + 1,
P~k (1,3,k .. 1,2). It is also
shown by Bose and Nair [,], that
The conoept of a strongly regular graph is isomorphic
with the concept of an association soheme with two associate olasses,
as intI'oduced by 1300e and ShilUamoto
of
part1all~I'
[4],
in connection with the theory
bni1a.nced desib'ns, if treatments are identified 'Iv! th
vertices, a pair of first aosooiatea with a pair of joined vertices,
and a pa.ir of soollnd associates with a pa.ir of unjoined vertices. Thus
stro(l(;:ly rCg'ular graphs fire t arose in oOl/neotion with the theory of
partiail:ly balanced designs.
The numbers v, n , n , P1jk are called the parameters
2
1
of the reGular graph G, They are oonnected by the relations
(2.1) -
(2.5),
chosen as
and only four are independent, TheBe may be conveniently
1
n 1 , n 2 , .P11
and
2
P11'
3, Pal:tial geoluetrics.
Consider two ~ndefined classes of objects oalled ROints
and lines, together with a relation inoidence,
fjUCil
that a point
and a line, ICla.y or ma.y not be incident. Th(.'Ill the yo1nts l:in1 lilH1S
aru enid to form a partial geometry (r,k,t) provided that the follo-
wing axioms are satisfied
I
A1. A pair of distinct pointa 16 not incident with more
thfin
Olla
line.
A2. J~ach point is incident wiUl \:lxl'lotly r lines,
A3, lJuoh 11110 1s incident with exu.ctl~ k pointl:l.
A,4. Given a point J? not incident with a line 1, tiH'Ire
are exactly t linea (t~1) whioh B.re incident rlith P, and aleo
incident with somo point incident with
1.
- 10 -
•
It there were two d1stinotlinee 1 and m eaoh inoident
with two dlgtiuot pointe P1 and Pi' then A1 would be contradicted.
Hence we ha.ve,
A'1. A pair ot distinct lin.s 18 not incident with more than
one point.
For convenience w. will use the ordinary geometrio language.
Thul it a point is incident with a line, .e shall lay that the point
liel on the line or is oontained ~n the line, nnd that the line
passes through the point. It two pointe are inoident with the I1ne,
then we .peak ot the lino as joining the two p01nt~. By A1 there
onnnot be more than one line joining two points. Thus either two
points are unJoined or Joined by a unique line. It two lines are in'1dent with a common point, they. are said to interseot, and the oommon
point is said to be their point ot lnt.~.eotlon. By A'1 two lines
oannot intersect in more than one point. Henoe two lines are either
non-1nt,rsooting or intersect in a unique point.
Theorem
Given a partial geometry (r,k,t) there exist
4 dual partial geometry PG(k,r,t) obtained by oalling the points ot
the t~r8t, the lines" otthe aecond. and the linos at the first
the pointe at the second.
~hi8 tollow8 by noting the dU8.11 t7 ot A1 and A' 1, the duali i7
A2 and A' and the self dual nature ot A4. In.taot A4 oan be rephrased
,.1.
.e
A'4. Given a line l ' not1ncldent with a point P there
are exaotly t points which are inoident with 1 and also inoident
wlth lome line incident with P.
In terms or the alternative geometric laneuaga introduoed
.e may write A4 and A'4 as
A4. Throut:~h any point P not lying on a line 1 thElre
pasa exac"Uy t linea interseoting 1.
At 4. On any line 1 not paardng through a point P, there
lie exactly t pointe, Joined to P.
The equivalenoe ot A4 and A'4 18 now obviou8.
- 11 -
•
4. GrAph or a partial geometn.
Tho graph
G ot a partial geometry (r,k,t) is defined
as follow8 I The vertioes ot G are the pointe ot the partial
geometry. Two vertices of G are Joined (adjaoent) 1t the oorrespending points ot the geometry are joined (incident with the atune
line ). Two vertioes at G are unjoined (non-adJaoent) it the oorresponding points of the partial geometry are unjoined (i ••• there exists
no line incident with both the points).
'l'heorem 4.1. Tho graph G or partial geometry (r,k,t) ia
strongly·regular with parameters
1~ t
fEr,
Let there be v points and b lines in the partial
geometry. Since the points ot the geometry have been identified with
the vertices or the graph a, we oan oall two points at the geometry
first associates if they are joined by a line, and second assooiates
it they are not jo1ne~by a line. Now through any point P of the
geometry there !,a08 l' ~ines, eaoh of whioh oontains k-1 other
points basides P. Henoe P has exaotly 1'(k-1) first Associates.
Henoe
n 1 • r(k-1).
Th1u shows that G is a regular graph. Conoider the b-r lines
not passing through P. p1rom A4 eaoh of those lines contains exaotly
, first ausoc1atee of P. 'Any partioular first assooiate Q of P,
li8s on 1"-1 auch lines, sinoo one of the 1" lines passing through
Q joins it to P. Henoe the number of first assooiate. ia
- 12 -
•
.A8'aln e8ch or the b-r lines not passing through P
contains exaotly k-t second ansooiates ot P. Any partieular
second associate R ot P ~iee on r 8uch line8. lienee the number
of second associates of
substituting for
b
P is
from (4.6) we have
Consider any two points P and Q. whioh are.1'lret associates. l'lloy are .joined by a line 1. We shall count the number of
points which are first assooiates to oaoh of P and Q. The k-2
points on 1 other than P and Q are first associates of both
P and Q• .Now there pasa (r-1) lines throuB'h P, other than 1.
By .44 eal.:h of these oontains t-1 tirst aS800iates ot Q, other
than P. lIhue these (t-1)(r-1) points are first associates ot
both j> and q.• I~ i8 easy to see that there are no other first
associates of both P and Q. lienoe
Consider two points P and R which are sucond associate8.
Thu4t is no line Joining them. Each of the r lines passing through
P ooutains t first aaaooiates ot R. lionce
2
P11 • rt.
Wa have now verified the formulae (4.1) and (4.2), sbowing
that G iu a etrongly regular graph. Tbe values of the other para...hr. Pjk tallow tro.. tho identities (2.,) and (2.4). The inaqualitT (4." olearly tolloWB trom axiom 44.
Corollary. The number of points
linea
b
y
and the nu.w"ber of
in a partial geometry (r.k,t) 18 given by
[(r-1)(k.1) +
t] /
t.
b • r [(r.1)(k.1) +
t] /
t.
y •
(4.10)
In view of
tllEt
k
ioomorjJhieM of
&88001al.101:1 sohel1u~s
with
two assooiat.e olasf'es, and strongly regular grsphu, the definition
of partially balunoed inoomplete block (PBIB) daeigns ~ivcn by Boee
and Shima-moto [4], may now bo rephraoed us tollow8 •
Giv~n a ut.I'ong"ly regula1' graph
G with parameters n • n ,
P~1' P~1' we may
1
2
vertices with v treatments. Then
aFBlll desi~n io anarranB'ement of the v treatments into b set.
(called blocks) such that.
('J;) flac!! tren. tmeu t is oon tained in exac tly r blocks
(p) Eaoh block oontains k distinot treatments
(r) Any two tr(~tlllents whion are first a.kl80oiatDB (joined
in G) ooourtogather"in eXll.otly A blocks. Any two treatments
1
whioh are second associates (unjoined in G) oocur togoth~r in A2 blooks
The dosign ma.y be oalled a 1'13113 desigll (r,k,A ,A ) based
identify its
v
1
on the 8 tl'ongl,)"
l'e~ular
2
gl'a.ph G.
Civul'l a pal'hal geollletrN (r,k,t), with gl'aph G, it is
olearl,y a l'Blli dosign (r.k, 1,0) based onG, and this 1:<B1.8 doaiB1l is
a oonneoted desiin. This tollows beoause two first assooiates alwaNB
ooour together in a blook, and if two treatments e
and 8 2 are
o 2
seoond aaaooiaiee, we oan find a trea.tmont 8
in P11. rt >0 ways,
1
and 8
al'O tirst Assooia.tQ8, and 6
and 8
are
suoil that 6
2
0
1
1
tirst assoo1atos. The incidenco matrix ot' a ,partial geome\.l'y may be
d..fined as the matrix .N. (n ) where n
• 1 ii' tbe 1-th point
ij
1.1
ia inoiden't with the j-th line and 0 otherwise. fL'llen N is alao
the incidence matrix of t.h~ cOI'roaporlciing FllIB doaigtu Now Conuor
and Clatworthy' [11] "and Bose and 1.188ner [2] , have shown that for
a oonneoted L'DII:l design, with two aooooiate 01a6008, liN' has only
three disti.uci 'oharaotari9tio roots, "hose lilul t1pilioit:lee are 1, Q;
and
"
where
- 14 -
",~
(4.12)
.
n
1
+ n
2
2
:+
(n - n g) + t(n 1+ n 2 )
_1
2
\T'1i'
,
and
r-
2
1
P12 - P12'
1)..
r2 + 2(1P12 + P122 ) + 1.
Bow theee multiploities are neoeosarily integral. Using the formulae
(4.1), (4.2), (2.~) and (2.4) we find that
•
Hence we have the 'theorem
~heorem
(4.2). A naoesoury
oondition for the existenoe ot
a partial geometry (r.k,t) io that the nUJllber
4·
rk
t
18 .. positive integer.
5.
Part1~111
balanced designs, whioh aro partial s-eometr'es.
\'1e have already Dhown that a partial geometry (r,k,t) is
isoll1orphio to a PJJIB design (r,k.1,O) based on the graph (al3sooia.tion
soheme) of the geoIIletry. llowevur a 1'131» design based on a strollBly
regular graph nuod not neoessa.rily be a partial geometry. It would
therefore ba of interest to find suU'!o1ent conditions under whioh
a PBID design (r,k,1,O) based ona strongly regular graph 1s ieomorphic to a partial geometry.
Mow Bou~ and Clatworthy [1] have ehown that 1£ there exist.
8. PBIB design (r,k,1.0) baaed on a strongly regular graph
G for
whioh r<. k,thon the parameters ot G are given by tho formulne
(4.1), (4.2) i.e. are the lIame as the pnrwnaters of the graph ot
80me partial geometry (r.k,t). We .hall ahow that the design, is
inde.d a plu-tit'!.l geometry and thUD eDtabli~Jh thetollowing theorem a
- 15 -
ThQoreIll(5.1). If thore exists a. FEID design (r,k,1,O)
basad 011 a tiltroIJ.gly regula.r graph G, then if r<k, the design must
be a purtial geometry (r,k, t) tor uome t~r. The parauwters of G
the formulae (4.1), (4.2).
The ptix·a.metero of G are given by (4.1), (4.2) in view of
the result of Bose and Clatworthy already oited. Also the axioms
are given
by
A1, A2, A~ for a partial goometry arc ixaxEtax implicit in the
definition of a f 1HB design (r,k,1,O). It therefore· only remains to
prove axiom A4 which W/lOUll.ts to saying that each blook of the deaign
oontainsaxaotly t troatments whioh are firat aaaooiates of a
given treat~ueu.t not contained in the blook •
. Let K bo tho set of k treatments cuntained in a partiou~ar block, and let R be the set of the remaining v-k treatments.
Let g(x) denote tho nWllber of treatments in R whioh have exaotly
x first aS6(lOiates in K. '£!lon
1
•
!-
g(x). v - k • k (k-1 )(r-1) / t.
x-O
(p,Q),
whero P is a
treat1l1ent in K, Q is a troa tmllnt in R , a.ud :P and Q are first
associates. Now e~ch treatment in K haD k-1 firat aS80ciates in
K, and consequently n ·k+1 first associates in !. Henoe the roqui1
red number is k (n,-k+1). Again there a.re g(x) treatluents ill Ie,
whioh llUVO exaotly x treatments in K. These treatments oontribute
xg(x) to our count. Henoe
Let us count the nwnber of pairs
Aea.in let ue oouut the number of trivlcta (P l'2,Q) where
1
P1 P2 is an ordered pair of distinot troatmentu in K, and Q is
a trar...tlllont in I whioh is a tiI'st assooia.te of both P1 al'ld P2.
Sinoe P1 and 112 havQ k-2 oonuuon .firs t aasooia tee in K, tl~ey
have P~1'" k + 2 oouunon first associates in R. Henoo the rtlquired
number of triplets like (P1 P2,Q) is k (k-1)(P~1-k+2). Now t1aoh of
the 8(X) trcutments in X, which have x first RU30c!atue in K
oontribute x (x-1) 8 (x) to our oount. Honce we have the equation
- 16 -
k
2C
1
x(x-1) g(x) - k(k-1)(P11-k+2) • k(k-1)(t-1)(r-1).
x-o
Uains (5.1), (5.2)
and
i •
(5.')
L
1.e. the averae~ value of
or
~lY
treatment
or
a simplo oaloulation shows that
xg(x) /
L. g(x)
- t,
x (the'number of first dasooiates ln
K
g 10 t. Alao
k
L:
.
g(x) (x-t)
2
• 0,
x-O
wh10h shows that
theorem •
•
x
must always have the value
t. This proves our
6. GeolIlotrisablo and psoudo-t)oornetrio urapha.
A strongly reb"Ulargraph G whioh has pnralnoters (4.1)
and (4.2) and for wl'.ioh tho inequality (4.') is aatiafied, is
defined to be a Rs<mdo-«eoemtrio graph with oharaotoristio8 (r,k,t).
'.rhus a pseudo-geo.Gtrio graph with oharaoteristio., (r.k.t) hss the
same pnraluQtElrs as, the graph of a partial geometry (r,k,t). However
a graph may be paoudo-e.:cometrl0 .1thout bolD£.! the gr£l.ph of Q. partial
geometry.
A 8ubtH! t of vertioes or a C;raph 0, any two of whioh Are
join~d 18 oalled a 01i9u8 of G. When G 18 the graph of a partial
~eometry there will exist in G'a set ~ ot diatingu10hed olique.,
1 , K , ••• , K , o01'r'caponding to the linea of the geometry
2
b
1
satisfying the following axioms a
A*1. Any two, Joined vertioes ot G are oontained in one
and only ono clique of ~ •
A*2. l'~aoh vurtex of G itl oontairwd in r o11quoo of L .
A*,. Eaoh olique ot Z 001l tE~infJ k verticfJR or G.
A*4. It F is a vertex of G not oon'ta1ned in a olique
Ie i ot L::. , thore are exactly t
vertioes in K whioh are Joined
1
to i ( i . 1, 2, •••• b).
- 17 ..
IL:mc(,i Bny gru..,ph C in which there exir; tu n set
Z
of cliques
1\1' K , ••• , ~, satiefyinc axioms A"N-1 - A*4 is the grH,;ih of a par2
tih.l Geometry (r,k, t). In fact G togetner wi·tIl tlUt cliques of L.
1s
i~0~orpJlic to
a
p~rtial geometry (r,k,t),
ponding to the pointa and oliquo6 of
2:
the vertices of G corresto the lines of tlHil geometry.
Such a graph will be eaid to be goometrisublo (r,k,t).
Oml ma.y considor graJilli3 in which there axis t a set
L.
of
K , K , ••• , ~ aa. tisfying one or mora but not all of the
2
1
axioms A*l, A*2, A*;, A*4, and investigato undor what u,id.itional
cli'lues
oondiUons the graph will ·be 6e01::llIltrisable. 'l'hue theorelJl (5.1) may
be rOllhraaed aa
'l'heoralU (6.1). If there exie ts a sot
••• , K
b
L
of cliques
in a. strongly re . ,ular graph G, oatisfying axioms
K
1
, K2 ,
A*1, A*2,
then G is geoL1~trisa.ble (r,k,t).
'rllOorem (6.2). Let G be a pseudo-geoiliotrio Gl·e.Vh witil
charaoteristios (r,k,t). If it 18 possible to find ill G a set
of
cliques K , K , ••• , K , satisfying axioms A*1 t"U!d ~\*2, and if
2
b
1
k :> r, tten G ia geoll1o"triaable (r,k, t).
Vie ai,all prove that oaoh of the oliQueo K , K , ••• ,
2
1
contains exactly i vertices, Bnd that if Q is any vertex not
oontained in any clique Xi (1 ~ i;i b), then th(~rc are exac tly t vertioes in K which D~O first aSHooiatca of (joinod to) Q. This will
i
show that if the vertices ot G are taken as points, and the cliques
K , X2 , ••• , K as linea. than wa have a partial geometry (r,k,t).
b
1
Let P be any vertex. Without 10s6 of genorality we aan
take K , K2 , ••• , Kr to oontain P. Now the sets K -P, K -P, ••••
2
1
1
Kr-P are disjoint and must ountain between thOlll all the r(k-1) first
aal30ciu t(:6 of 1>. :l!'roro this it follows that the average number of
A*, and if
k >
1',
L:
Ko
vertices in tho
r oliques of the Got Xl' 1 2 , ••• , K is
r
there exists a clique oontaining at leaut k verticos. Lot
such a cli'lue. Lot
exaotly
K. Lot
k
g(x)
Ull
ta.ke a Dubeet
vortioes. Let
1
R be tbo set
of
K , such that
j
k. Hence
K bo
K
J
oontains
of vertiooa not contained in
be tho number of vortices in
R whioh
have exaotly
tiret a060ciates in K. Then it tollows exaotly as in tbe proof ot
theorem (5.1) that
l
x
- 10 -
k
L:
g(x) • k(k-1 )(r-1) / t,
x-O
and
k
L
xg(x) • k(r.. 1)(k-1).
x-O
Bence tho average value of
k
~
x
2
is
~.
(x-t) g(x).
Alao aa before
0,
x-O
•
.hieb is only ponsible if x is eonl3tant and aqual to t. lIenoe
every point of R has exaotly t first asooointea in K.
If K
oontains any vertex Q other than thODe alrf;lHdy
j
contained in 1<, . then (~ belongs to X, and therefore has eXB.otly
t first asaoeiat~s in K. But eaoh point of 'K Is 11 first aooooiate
ot Q henoe t-k, whioh oontradiots11it ;ir<k. 'l'hie shows thnt
none or tho oliquQS K oontaining P, oontains more than k vertioe••
j
ThUD eaoh oontains exaotly k ~ertioee. Sinoe eaoh olique oontains
at least one vertex eaoh of the olique K , K , ••• , ~ conts.ins
2
1
exaotly k vortioes. Aleo 1f Q is a point not oontained in
Xl (11i'i ;jib), then thero are exactly t vertioes in Ki which
are first a8~ooiat&s of Qt' This oompletes the proof of the theorem.
N.D. Compare theorell1s (6.1) and (6.2).
7.
Exnqplusof partial geometries.
(a) A net (r,k) of degree r and order k 18 a system
ot undofined points and lines together with an incidenoe relation
subject to the following' axioms (i) There is at least one point
(ii) The linea ot the net con be partitioned into r disjoint,
non-empty, "perallel olasaee » such that each point of the net is inoident with 8XtlOtly one line of eaoh ulnae and given two lines belonging to diatlnot classes there i8 exaotly one point of the net
whioh is inoident with both lineae
- 19 -
}'or convenience we eHn
tH'JfJ
phrasos such as Itpoint in on n
linelt is toad of speoking of incidence. Thon it CBn be readily prayed
(Dce for exuuple Bruck
[5] )
that
(1) Euch line of the net oontains exaotly
pain ts whore
k
k
distinot
r
distinot
~ 1.
(2) Baoh point of the net lioa on ex~\ct1y
lines where r S:1.
(3) Tha net has exaotly rk diot1nct linos. These linea
fall into r parallel classos of k lines alitch. D1Btinct lines o.t'
the D[~lilO. parallel claGO have no oomrr,on pointo. 'llwo linen of differont
classes hove one oommon pblnt.
(4) The net has exactly k 2 distinct pointa.
vre shELll ahow
thllt ~net (r,ld of deGrcQ
r ' and order
k
1e a purtinl tieolilatry (r,k,r-1). ~lha properties (1) and (2) a.boYe allow
•
that axioms A' and A2 of a partial geometry hold.
Twolinos oannot 1nte~seot in nora than ono point, for
they 01 ther bvlong' to the onme parallel olass and ha.ve no common
point, or diffoI'ont p,Hnllel ola.ssos in which they havo onl) OOInl,lOn
point.
~rom
this follows the tact that two distinct points cannot be
inoident with illore than one line. Hence a.xiom A1 for a. pll.rtinl gfiOmetrt holds.
Again given a point P not incident with the line 1,
'l'here a.re exactly ,r
olass.
lines
throU~fh
P, one belon[;'ing to llach jJruAllel
1 (i.e. b(!longo to tho emllo
1), Elnd does not intersect 1. 'rho other r-1
une ofthcae io pnrallel to
parallel
ClllSU
lines tlu,'ouCh
as
P
eaoh intersects
all dietinct. Honce axiom
1
in one point, thesa points being
A4 for a partial geometry holds with
t • r-1. This completes the proof of our statement.
I·t f0110wo from thooZ'OIU 4.1, tha.t tho pvrameto:cs of tlle
graph
ON
of' n net
(r,k) uru give'I! by
n 1 • r (k-1),
P~1
ti 2
• (r-2)(r-1)+(k-2),
• (k-1)(k-r+1),
2
P11
- r (r-1).
- 20 -
It a etrontI1y regular graph haa plJ,rarneters (7.1), (7.2) .e
shall call it a pseudo-net graph with oharacteriaticB (r,k). A
po~udo-net.~raph with charaoteristios (r,k) is pseudo-geometrio with
oharacteristios (r,k,r-1).
Druok [5], defines the deficienoy
of a nat (r,k) by
d
d • k - r + 1.
•
The inter~retat1on of the defioienoy d is that if it
were possiblo to add d more parallel olasoos, eaoh oonsisting at
k 11neo, so that the extended net now has k+1 claases of parallels,
th" net l~ould become an affine plano, in whioh any two points are
Joined by a unique line.
If_e take the k 2 points of the llet as troatmontB and the
rk linos aD blooks, we obtain what 1a knoW11 aa the lattioe deaign.
This 1s a PJ3!,B design (r,k, 1,0)' based 011 tho utrongly rogula.r graph
GN with paramoters given by (7.1), (7.2). Lattioe designa were introduoed by Yates [2~. Tho af)~ooiation scheme oorrosponding toGN 1.
the L soheme defined by Dose and Sh1mamoto [4].
1'
It 1s well known that a lattioe desi~~ with r replications
and blooksi2e k is equivalent to a sYiltom at 1'-2 mutually orthogonal Latin equares at order k.
If 1'-2 'mutuall;y- orthogonal· Latin equates of order k are
given, we oan superpose them. 'l'hon each 0011 contains 1'-2 symbols
belo~ging in order to the different Latin squares. The k2 cella
are now idontified with k 2 troatments, Treatments belon~ing to tho
smae row e;ive ono uet of k bl.ooks. Treatments bolon6'ing to the
Bame oolumn give another set of k· blooks. Treatments (oella) whioh
oontain tho aUlae Bymbol of the i-th Latin square give a s.t of blooks
tor each value ot 1 (1 • 1, 2, •• e, 1'-2). We thus Bot l' seta of
blooks. The treatments and blooks 80 obtainetl oonstitute a lattioe
deoign.
Conversely given a Lattice des1(&D. witb
blook 81z8 k. we oan oon8truot a .et of
Latin squal'eo ot o,rcler k.
r-2
r replioations and
mutually orthugonal
.. 21 ..
It
r-2
mutt.<all:{ orthog'onal La tin equates of ordor
k
are
given, we can superpose them. Thon eych oell oonta1na r-2 symbols
2 oe110
belonGing in order to the different Latin squares. The k
are now identified with k 2 treatments. Treatments belonging to the
same row give one set ot k blocks. Treatments beolonging to the
e'ante oolumn gi va another eet of k blocks. Tree. t!nonts (cells) which
oontain the CHiLIEl symbol of tho i-th Latin square give EI eet of blook.
for eaoh vnlue of i (1.1,2, ••• , r ...2).We thUG 'get r Beta or
blocks. ~'ho tren·te·Lents and blooks eo obtained oonati tute a lattice
a Lattice decign with r replioations
k, we can construct a set of r-2 mutually orthogo-
Conversely
and blocl\:P' size
~iven
nal La tin aq.uures of order
k.
n Xn
(b) Take an
equares and write down the numbers
in tho 0(1110 above tho main dalli?:onal. Fill up
1, 2, ••• , n(n-1 )/2
the oells below the main dif.t~onR.l syuufictrioally. The oase
n· 5
i8 oxemplified below.
1*
, '..
1
12
·'·T·· .,. r"'···
.
~T"'
':4
..··--·T-·_·_ . i
\11*15\617\
1;
',''\'6
\ "2
I
;
!4
j'* ~ e
(
i
T9 i
!
!··~·····r~····T"·~(;1
I
r· ··· ·+·..".._··. "1-····,,··1'.. · ··1
I 7 1. 9.L10 J. * J
Fig. 1
Tile oe11s oontainiHg' the same nUluhQr aro identifies with
the sllr.e point. 'l'hua tbere are two different cells represonting the
pointkJ 0.11 togethor. Let the
nrowu oouflti·t;ute lineD. ThuH there a.re n linua. It is OJ.'H.1.);, that
axioms A1, A2, A3 of a partial t:"Oolllf~try are sa. U&1'ied wi tb. r . 2,
k • n-1. It itl eaay to Bee that any two lines 1nt.rsoot in one;;, point.
Thus t.. 2, aud we hllve a partial geometry (2 t n-l, 2).
If twu points whioh liQ on a line are called firet assooiat •• , (:Ind two pointe w.niotl do not lie on a~ linu are galled seoond
8arue yOillL, thttro being
v . n(n-1)/2
.. 22 -
aueociatos, we have tha trio.n(;"Ular a$sOciatiOll scheme first defined
by Dono and ShiltlHraoto [4], and extensivoly studie.:t by Connor [9],
Shr1khande (21], lIo.rfr:lan [13,14] and Qumg [6,7J. The purarlletera of
the assooiation schemo or the o~)rreaponding strongly regular l;,'Tflph
o.re
n 1 • 2(n-2).
pl1 • n-2,
trongly regular gra.ph has the paramf} tero (7.4), (7.5)
we 8hall call ,. a pseudo-triangular graph with,oharaoteridtio n.
If a
8
A pseudo-tria.n{;ular graph with chs&'aoteristio
with charaoteriotioB (2,
n
is 1)81.'JUdo-£;eometric
n.1, 2).
(0) A balanoed inoolltplatu block design DIll is an arrangement
of a set of Yo objeots or treatments in b
eote or blooks, such
o
that (i) oaoh blook contains k
distinct troatments (11) eaoh
o
treatment is oontained in r o blocks (11i) each pair ot distinc*
treatments is contained in A blooks. This design has eomctimoD
o
b(len oDlled a (v0' ko,A ) conflgurntion. The dunl of Ii design 18
o
defined as a new design whose treatments and blocks are in (1,1)
oorrespondenoew1 th the blooks and trae, tillenta of the original design,
and inoidenoe 1s p:teserved (Where a blook and. a troatrnent arEi inoi-
dent
if'
the treatment is contained in the block, and non-incident
otherwise) ,Shrikhande
[2'1
has ohoYfn that the dual of
It
llID design
with
A • 1 is Ii\. FBID design. Now a BIB design with A • t is
o
0
olearly a partial BeOtUetry (ro ' k o ' k o ). lIenee the dual design is tlle
dual p~rtial goometry PG (k , r , k ). It we eet k • rand
000
o
the replioation number and
the block
e1z. irA the dunl desib"l1. then the dual dasign io the partial geometry
(r,k.r). Any t\90 blocks (11nos) of this design intersoot in Ii\. unique
treatlllerlt (point). Hence the osuociation schomo of this dODign has
been oalled the SLD (singly linkud bloak) uoheme by Dosu and Shima140tO [4]. !l!hu pu'ultloters ai' the oorr00pon/1!ng strollgly rogulnr graph
Qan be written down direotly from theorem (4.1). We be,tVe
r o • k, so as to fna.ke
r
k
n2 • (k-r)(r-1)(k-1)/r,
n 1 • r(k-1),
p~ 1
+ k - 2,
• (r_1)2
2
2
P11 • r •
It a strongly regular graph haa the parameters (7.6), (1.7)
w.e shall oall it a pseudo-SL:B graph with oha.raotor1~t1c8 (r,k). A
pseudo-SLB graph with oharacteristios (r,k) is a pseudo-geometric
graph with characteristics (r,k,r~.
(d) To oonolude we shall give a rather les8 obvious example
ot a partial geometry.
•
Consider an elliptic non-clegenerate quadrio Q in the
5
f1nite proJeotive spaoe pG(5,pn). This quadric is ruled by straight
11nes, called generators, but oontains no plane. As shown by Primrose
[16] and Ray-Oha"dlluri (17), the)te are (.'+1)(s+1) points and
(.'+1)(.2+ 1) generators 1n Q5' eaoh generator oontains s+1 points,
2
'n
and through eaoh point there pass • +1 generators, where s . p •
It P is a point on
not contained in a generator 1,
then the polar 4-space of P intersects 1 in a single point P*,
and iP* i. a generator ot
It oan be readily verified by using
theorems provea by Ray-ChnUdhuri that iF* i8 the only generator
through i. whioh interseota 1. '.L'his showl3 that it we oonsider the
points and generat9rs of
as points and linos, they oonstitute
a partial geometry (.2+ 1 , .+1, 1). The parameters of the graph ot
this partial geoluetry oan be easily written down using theorem (4.1).
They are
Q,
Q"
Q,
n
1
• a(.2+1),
1
P11 • 8-1,
The partial geometry (8 2 +1, 8+1, 1) is ot oouree a PDIB
design. Thiu was obtained by Ray-ChaUdhuri [1SJ, the speoial on·ee
a • 2 was given earlier by BOBe and Clatworth~[1]. An intereoting
,
point in tbe present formulation 1., that to ver,.fy that the oonfiguration of pointe and generators on Q 1s & PBIB design we have only
5
to cheok the oonst.ano) ot r, k and t instead 01' the oonotanoy ot
- 24 -
1
2
as was done by Dose and Clatworthy [1J and
2
by Ray-ChaKdhuri [10]. The dual partial geometry (.+1, 8 +1, 1) 18
alHo ot interest.
In the SIlIUQ way one oan show that the oonfiguration of
points and generators on a non-dogenerate quadrio Q in Pa(4,pn )
4
a partial geometry (8+1,8+1, 1).where 8 . pn. The oorresponding
design was first obtained by Clatworthy [a].
r, k, n 1 , n 2 , P11' P11
1s
8. Lemmas on olawe in pDeud(l-c~eoInetr1Q g-rar·hs.
shown in theorell1 ~5.1), that 1f we can base a PBIB
design (r,k.1,O) on a strongly regular graph a, then Q 1s pseudo...
geometrio and tho design is a partial geometry. Vie oan ask the oonverse question. If the gra.ph G 18 p8eudo-eeolM~trio (r,k,t) oan
we base a PDID design (r,k,1,O) on it? In graph theoretio language
this question may be put 88 I If the graph G 1s pseudo-geometrio
(r,k,t) oan we find a Bet otoliques K K2 , ••• , ~ satisfying
the axioms A*1 - A*4 of seotion 6.
"
In the re~t of tho paper we shall fre~ueJltly use tbe
tollowin~ functions ,
We have
(a.1)
f(r,t) •
(a.2)
q(r,t) • 1 + (r-1)(21'-1)(t-1),
(a.,)
f(r,il) • rt + (r-1)(t-1)(2r-1),
(8.4)
p(r,t) ~
1 + (1'_1)2
t [r (1'-1)
We nato that in view
(a.5)
per, t)
~
or
(t-1),
+ t
(r+1)(r2-2r+2~.
the inequa.lity
r(r, t)
1~t:::1'
~ q(r, t) ~r<r, t)
The ooncept of a claw waD eugffoeted by Alan Hottman in
oonversation wi th H.n. Bruok and the author. By a olaw [p,a] ot a
pseudo-geometrio grap.h G is moant an ordered pa.ir oonsisting ot
a vertex f, tho verts! of the olaw, a.nd a non-empty aet S of vertic••
diotin.ot trom I) such that overy vertex in 5 is joinod to P in a but
no two vertioes in 8 are joined in G.
- 25 -
The number of elements in a finite 8et 5 w1ll be denoted
by
lsi.
The order ot ille olaw
In Lemmas 8.1 - 6."
[P,s]
is defined as s · lsI.
G denotes a pseudo-geometrio graph
Lemma 6.1. It k ~t(r,t) • 1 + (t-1)(r-1)2 then tor any
8,
1~.,~r. eaoh vertex r or G is the vertex ot a ol14w· [p,s]
of order s. We can ohoose S to inolude any vertex A joined to P
in a.
Let P bo a vertex of G. Suppose tbere exists a olaw [p,s]
ot order· 8. Lot l' be the set ot all vertioes other tban those
belonging to [ptsJ and which are joined to P (aro first asso01ates
of p). LlI!t rex) be the number or verticos Q in T, such that Q
i. joined to exaotly x vertices in S ( rex) is the number ot verticos in T. eaoh or which has rex) first assooiate8 in S). Then
we have
(0.6)
•
~ t(x) • n, - •• r(k-1) - e,
x-O
sinoe the left hand 8ide of (8.6) oounts all rirst &08001ate8 ot P,
whioh are not in S. Now let uo count pairs (A,Q) where A 1. in S
and Q 1s in T and 1. a first assooiate of both A and P. Sinoe
1 oommon first. &88001ate8 ( none of whioh
A and P havo exaotly P,1
oan bolon~ to S by the definition ot a olaw) we have .p~, paire
l1ke (A,Q).
Again there are rex) vertio8a in T each of. whioh has
ex_otly X t~rut ausoc1ates in S, They contribute xr(x) to our
oount, lIenoe
(e,7)
Bonoe
(e,8)
e
teO) -
L:..
x.,
(x-1) rex)
III
(r-e)(k-1) - 8(t.1)(r-1).
- 26 -
Henoe it 8<r and k 7-r(r,t) • 1 + (r_1)2 (t-1), then
teO) is positive, i.e. there io at least one vertex Ao+1 in T
which io not joined to '1' A2 , ••• , As' We oan thorefore add As +1
to S and get a claw of order 8+1. In this way we oan go on extending a claw till we ge't a olaw ot any required oI'dar not. exceeding r.
We oan e tart thie 'prooea~ with any olaw [P,A] of order 1. This pI'ovel
the Lemma.
Leillma (0.2). It k >('(r,t) • 1 + (1'_1)2 (t-1), and if [p,s]
is a olaw of order r-1, thun thero 'oxiot at least k - (r,t)
distinot vertioes Q of G such that [p, SUQ] i8 a olaw of order r.
It [p,s] is a olaw of order 1'-1, then from (0.:5)
t(o)~
k-r(r,t).
Henoe thera exist at least k - r(r, t) vertioos Q, eaoh
of Which taken toe;ether with S give a claw [p, s*] of ordor r
wnere S*. au'l.
Lemma (a.3). It k:;.. per, t) •
[1'(1'-0 + t(r+1 )(1'2"'21'+2)]
then there exists in . G no claw of order r+1.
Let ~~,s] be a olaw of order 8 in G. Let the set 'J.I
be as in Lemma (0,1). We shall oount the number of triplets
(A,A2 ,Q) where '1' '2 is an ordered pair otdist1not vertioes in
S. and Q. i8 a vertex in '1' whioh i8 a tirst aSfSooiate of both
.1 and A2 , Sinoe '1 and '2 are seoond ae800ia1.8' 1.hey have exaot2
11' '1,-1 oommon tirot aa'Sooia tee othel' than P, Some of these lllay
not lie in T, Henoe an upper bound tor tIle required number of triplet,
1, _(e-1) P~1' However the t(x) vertices in T, eaoh ot whioh haa
exaotly x first associates in S, oontribute x(x-1) r(x) to our
oount. lIenoe
t
..y x
x.o
It
(x-1) rex)
~
•
(a-1)(p~1-1)
• a (s-1)(rt-1).
possible lot s . r+1. Then adding (0,8) to (8.,)
multiplied by a halt, and noting that on the lett hand uide ot (8.9)
t11e term x. 0 contributes nothing wo get
.. 21 ..
L•
f( 0) + ~
(x.. 1 )(x-2) f(x)
x-1
~­
::=
k > per, t). there l)8lU1ot exist a olaw of ordor
Helloe if
r+1, as the
left hand aide ie eS8entially pouit1ve.
9. Lemmas on oliques in pS9udo-geoUletrioEaphg.
In the Lel1Ullas (9.1) to (9.6), G denotes a pseudo-geometrio
graph (r,k.t). Tho definition ot a major clique generalizes Druok' •
•
definition. and the oonoept of a grand olique 48 taken over from
c
:Bruok
(5).
K ot
A major clique
l1(,1 ~k+1
-
fer. t)
G is a olique suoh that
• k - (r_1)2 (t-1).
A Brand olique is a major olique which is also a maxima.l
c11Q.ue.
If
K
ar(l <Uatinot maximal oliques, thEm
and. L
KUL·
oannot be a oltque. Sinoe K and L are diotinct. there must be
a vertex P in one of them (aay K). not belonising to the other (L).
Now if KUL is a olique, then P is joined to every vertex ot L.
Thus . pilL is a olique which oontrl\dicts the fa.ot that L is maximal.
Since grand oliques are maximal the union of two·grand.oliu,ueo cannot
\
be a 0119ue.
t.
If we take the set of grand oli'Lues as t.hu set
of seotion
6, we may enquire under what condi tiona the flxioIIlS A*1. A*2. A*3 and
A*4 are satiatied. The LemJitsa which follow. ara direoted to this
purpose.
Lemma (9.1). It k >- (r, t) and if Q h~B no ole.w of order
1'+1, tlHm for every pair 01' diatinot joined vertioes P and Q in
G., there exists at least one .aJor olique oontoinins both .Ii' a.nd Q.
i1rom Lemma (0.1) we oan fi¥ld a olaw. [p,n] of order r
luoh that Q6 S. Let A1 , '2' •••• 'r-1 beothor vertioes of S.
Let
be the set of vertioes R whioh when aliJoinod to S, ~1V8 a '
olaw IJ,S* of order r. $*. sUn. Prom Lerrane. (0.2). the number of
n
points in
a
-/.01 ~ k
-
r
(r, t).
- 28 -
Tho vtn'tioee in
n
are ull joined to one another. If any
two wero not joinod they could be added to
give a olaw of order
P
r+1. Thus
pUn.
mutuall,y joined. Henoe
A • A ••••• A _
to
1
r 1
2
and the vertioeo in 12 are all
K
i;'k+1 - ( r , t),
is a olique of order
i.o. a major olique.
Cor 1. In Lemma (9.1) tho hypothesis may be replaoed by
>~ (r. t).
k
This tollows from Lemma ,(a.,) by noting that
for
p ('b,t)~t(r,t)
1 =i t ~r.
Cor 2.
tist1ad, P
Whon .he oonditions of'
(~
and
L(:)nll11a
(9.1) or Cor. 1 are sa.-
are oontainod in at least one grand clique.
We oan extend the major olique K by adtiinc new vertices
till it becomes maximal, and therefore a I;;rand clique.'
Lemma (9.2). If K and L are cli(lUOS 01' G and KIlL 1s
not a olique then
Sinoe
of vertioes
Ix n L I~ rtf
KilL
is not a olique, ther£!
n~t
.,"2
tilllSt
exist in
joined to one another such that
'K () L'
a pair
KUL
P,~
K,
'2e L. Any vertex belonging to
mua t be joined to both
and P2' lienee the number of veJ'tioes in
n L oannot exceed
Ix I
2
P1
P11. Thus
Lemma
(9.3).
If
K
~ld
oontains at least two vertioes
Every vertex in
both
A and
KUL,
L are oliques of
A and
other than
Q
and
KnL
B, then
A and
B, is a first assooiate
ot
B. Henoe
IKdLI ~
Lemma (9.4). If
P~1 +
2 • k + (r-1)(t-1).
K and
(1) KtIL is not a oli(lUe (1i)
then
xn),
I K I + IL I ~
L
are oliques
ot G such that
oontains at least two vertioes,
k + rt + (r-1)( t-1 ) •
.. 29 ..
~ino. oonditions of Lemmas
(9.2) and (9.') are satisfied
:;rt + k + (r-1)(t-1).
Lemma (9.5). It k>p(r.t). rt + (r-1)(t-1)(2r-1) and if
a ha~ no olaw ot order r+1. then two distinot vertioes ofG which
a:n joined, are contained in one and only one grand olique.
Suppose there are two distinot grand cliques K and L
both oontaining }> and Q. Then KUL oarUiot be a olique. Aleo KnL
haa at least two vertices P and Q. lIenee from LelilUIB t9. 4).
(II + (LI
Sinoe
K
and
L
2 [k -
~rt .+ k + (r-1)(t-l).
are grand oliques. they are both ma.jor. lIence
(r_1)2 (t-1)] if
Ixl
+
IL' ~
k ~rt + (r.. l)( t .. 1 )(2r-1) •
rt + It + (r-1)(t-1).
p (t).
whioh is a cont.radiction.
Oar. In Lelluna (9.;) t.he hypothesio may be replaoed by
It > ~ (r, t).
Lemma (9.6) It (i) k>q (r,t) • 1 + (r-1)(2r-1)(t-1),
(li) two distinot vertioes ot G are oontained in utmost one grand
olique at 0, (iii) the.e exists no claw of order r+1 in 0, then
each point at a 1s oontained ,in exaotly r grand oliques.
From Lemma (9.t), Cor. 2 any two vertioes are oontained in
at leaat one grand oli'jue. It follows from assumption (il), that any
two vertioes ot G are oontained in one and only one grand olique.
Again from Lemma (0.1): there exists a claw [p,sJ of ardor r, where
e • {A 1 , A2 , •••• Arl' As
T be the oet of tirst
associates at P, other than .1 ' A , ••• , Ar~ We 40fine H as the
2
j
1
.et cons1et.ing of P, A
and
all Q T, Buch that Q io a firs t
j
&6oociate ot A j .but not ot Ai when i I- j. L e t t (x) be as in
in Lemma (0.1), let
e
LemVla (0.1). Now
reo),
01n06 there are no claws of order
r+1, llanoe
from (8~5) and (0.6) we have
r
L
x-1
r
~
f(x). n l - r • r(k-2),
xr(x) • (x-1 )t(x) • r(k-2) + r(r... 1) (t-1).
x-1
r
~ (x-1) f(x) • r(r... l)(t-1)
x-2
Now
r
~f(x)
2
r
r( t-1) :S ~t(x) ~ r (r-1)( t-l).
2
r (t-1) S r(k-2) - t(1)
~
r(r-1)(t-1).
Henoe we have
two
vertices in H are joined together otherwise there
J
would be a claw ot order r+1. Thus H is a clique.
J
It we put Hj. Hj ... (AjUP), then
oonsists of exaotly
those vertices of T which are joined to Aj but to no other vertex
of
Henoe Ill, H~, ••••
are disjoint seta and the total number
ot vertioes in theae seta is f(1), whioh sutisfies (9.1).
Uow there 1s fl uniqu.e grand olique K
oontaining A j
J
and P (.1. 1, 2, ••• , r). The number of vertioes in K oannot
J
be les8 tharl the number or vertioes in H • It };lo6sible let
J
IlCjl</HJ/. Sinoe Kj is fl h'Tand 011quo it folloW8 that Hj 1s a
major Clique and is o0utained in aome grand olique Kjt Sinoe A
J
and P al'e oontained in Kj
and Kj' they must ooinoide •. Honoe K
J
eontains Hj whioh oontradiote IKJI<IuJI.
An.y
Hl
s.
a;
- 31 -
consider the
K ••••• K ,
2
r
Ki-i)
and
Kj-P
Then K, -P, K2 -P, ••• , Kr-P are disjoint. Ji'or if
"
(1 ~ .1) have a OOUlIllon point (~. then K and Kj would ooincide and
~ow
grand cliques
r
K
i
would contain both
joined to A • How
Ai
Bnd
Aj
which in impossiblo since Ai
io not
j
• r + r( 1)
If pogsible ~up'poae there 18 another grand clique Kr +1
oontaining P. The vertioes in Kr + ·P must be disjoint from the
1
vertioes in K, -P, •••• Kr-P. Also K +1-P must have at leaat
r
(k-1) - (r.,)2 (t-1) vertices, If we remember that the number of
first associates of P is r(k-1) we have
k~ 1
+
(r-1)(2r-1)(t-1).
k ~ q(r, t), whicb g1veo a ountrad!c tiona This finally proves
lIenee
our Lemma.
(1)
Cor. 1. The hypothesis of the Lamma ma.y be replaoed by
k >f(r,t) and (il) th~ro exiatu no olaw of order r+1 1n
Il'his followo fI'Olll Lemma (9.5), anel thE) lnaqual i ty
r (r,t)
~
q(r,t).
Cor. 2. Tho Lypothesia of tho Lemma may be replaced by
k >
p(r,t),
. fj,lldo
per, t) ~ fer, t).
follows from Cor. 1,
Lell'lUB.
(8.3),
1l1'ld
the inequa11 ty
G.
Theorem (9.1). Let
G' be a paoudo-goometric gra.ph (r,k,t).
If (i) k>q(r,t), (i1) two distinot vertices ot G are contained
in utmost one gra.nd clique, (1ii) there exists no ola.w ()f order r+1
in Gf then G is geometriaable (r,k,t).
~heorem (9.2). Let G be a pseudo-geometrio graph (r,k,t).
It (i) k >p(r,t), (11) there flx18t& no claw of' order r+1 1n G I
then a i~ geometrisable (r.k,t).
Theorem (9.3). L.t G be a pseudo-geometrio graph (r,k,t).
It k >p(r,t), then 0 10 goometrisa'ble (r,k,t).
If we take the Bui of grand oliques of tit as the oet
at section 6, them LemIllas (9.5) and (9.6), together with tlw1r oo~la­
ries I$how. that the axioms A*1 and A*2 are uatiefied, \mder tho
oonditione of fl.ny of the TheOl'ems (9.1), (9.2), (9.]). '1'110 rODult
now follows from Thoorem (6.2).
- }~ tor n>8. This result was first obtained by Connor [9]. or oourse
when we use design of experiments language the vert!coo or the graph
or points are truatments.
Shrlkhande [21] hss proved the uniqueness of the triangular
soheme for n . 5,6 and Hofrma.n [,,) and 6heng [6] have proved the
same for n - 7. Both Hofrlnan [14] and Chang [7] have shown that for
n -8, the parameters (1.4), (7.5) do not completely doteruine the
soheme. 'Thore are three. other possible schemea with the same parameters beeideo the trianb~lf.l.r. Thi' may be expressed &s follows r
:rhere are four non-isomorphic 8 trongll resular graEhs wi th pCl.rftI'Jeters
only one of whioh in geometrisable
(2,7,2).
Oonsider a BID design
v* •
t (n-1~(n-2),
b*.
t n(n-1),
1'*. n,
k*. n-2, A* - a
Hall and Connor [12] have shown that it this design ex1sts then it
Gan be embedded
i~a
symm,trio BIB design
ro •
k
0
•
n,
A • 2
o
Their proof dOEls not OOver the oase n • B, tor whioh Oonnor [10]
••perately showed that the design (9.1) doos not exist.
ihrikhando [22J, haa .proved the Hs,ll-Oonllor theorem tor the
ca.. n ~ 6 by using the uniqueness ot the triangular soheme for
n ~ 8. It is intereetin~ to oblJerve that n - 6, the (Jase not oovered
in Hall and Connor t • entirely different proof, i8 exaotly the oae.
when the parameters (7.4). (7.5) do not uniquely oharaoterise the
80heme 'UI triangular.
t1.
(l'heorems of Shrithande r Druok and
M~sner
on tho
uniqueness of the
(r,k)
(1.2).
Lr, scheme.
~onDider G
as a pseudo-net graph with oharaoteristics
N
or the corresi/onding auaociatiori sohema with parameters (7.1),
Since t . r-1 in this oase, GN is geol!letrioa.ble (r,k,r-1)
it
k >p(r,r-1) •
t
(r-1)(r' - r
2
+ r + 2).
In partioular if r . 2, this reduoes to k>4.
In the oase r . 2, the geometry oonoiets of two sets ot
parallel lines. Each parallel 01a88 oontains k lines, and e~ch line
oontains k points. Lineo of the sarno 01as8 do not interseot. Lines
ot different classos interseot in a point. Thus eaoh point is uniquely
determined as the interseotion ot one line from eaoh clasa. We oan
number the lines of eaoh olass . 1, 2, ••• , k, and we oan number
the points or vertices ot the graph 1, 2, ••• , k2 • We now take a
k x k square and identity the i-th line ot the first class with the
i-th row, ths j-th line ot the seoond-class with the j-th oolumn,
and the cell (i,j) with the point which ia interseotion ot the line
t of the tirst 01a08, with the line j ot the second olaDB, then
the aa8oci~tion relations between the verticos ot the graph are exhibi ted as an L Boherlle. Thif. proves the uniqueness of the La scheme
2
tOl'
n >4 a result obtained by Shrikhnnde (19]' and by Mesner [15].
In the general oal3e the geometry oonsists at a set of k 2.
points (tho vertices ot ON) and a r clasoes at parallel linea,
eaoh olass con'taining k linos. Lines at the sa.me claao do not inter.eot. Lines of different cla.Elses intersect in one }loint. Let the
parallel olasaee be designated by (R). (a), (U ) ••••• (U r _2 ). To the
1
k linos within eaoh 01a88 we assign the symbols 1, 2, •• 4\, k. Ea.ch
point is uniquely given by the tntereoction ot a line ot (R), with
a line ot (0). Hence as in the oase r . 2, the linos of (R) may be
identified with the rows, and the lines ot (0) of tho columns of a
k x k square. Then the intersection ot the line ot (R) a.nd the line
ot (0) 1. identified with the oell (1.J). It in eaoh cell (i.J) we
put the number or the lin. ot (U(,I) whioh paasee through the point oorr •• p,onding to the cell, we get a Latin uqut'lre" L_
'"
(It. 1. 2, ' ' ' ' r-2)
- 35 -
and the Latin squa.re13
L
L , ••• , L .. a.re rnutuall;y orthogonal. 1'wo
r 2
2
points (cella) ar<:l first " u13socia.tee if they lie in the some row, same
oolumn or oorrcopond to the same letter
or
the
BaJllEl
La tin square.
Thun the auuociation rclaUons between the points or vel'ticea ot . G
N
oo.n be exhibited by the L
eohume defined. boY Boae a.nd ~himarnoto [4].
r
Thus the L
scheme with parameters (7.1) .. (1.2) 10 unique (up to
r
type) it (1'1.1) holds. It ia lHiOOU£Hl.ry to £l.dd the words up to tYl)e,
since thal'E! ma,y be l.nany non-isomorphic setu of
r ..2
lAutually ortbo..
gonal Latin squares. 'llhia result ~ _ is
implicit in 'Bruok's paper
"
[5]. A s-tt,jM~ ~~. AA...~ ~ ~
NU~ f IS''].
12. The SLB sohElme and· the goneral uniqueness theorem.
Let oonsider the SLB oohem8 or the pseuclo.. SLB Graph with
cha.racteristios (r,k) for whioh the parameters are given by
(1.7). Then
thcOrQfU
(7.6),
(9.3) states that the graph ia geot!etriaable
(r,k,r) it
In thtl language of designs tiliu would lntlan tho. t ii' there is an asso-
ciation scheme with parameters (7.6), (7.1) then it (12.1) holds the
asaooiation r~lationo oan be exhibited by the dual of a ~IB duaign
(With r o • k an~ ko • r. Ao • 1) 80 that the first associates are
exaotly those widoh ooour together in a block of this dual and the
leaond &00001at8$ ar8 those whioh do not occur together in a blook
or
this dual. Thus
(7.6). (1.7)
oiation £lohome up to type. It is
determino the llh'uoturo of the assoneoe6~:Hlry
to add the wordo up to
type lIinoe there will exist in general non-isomorllhia DIB d,esit\118 w1 th
(r o • k. k o • r, Ao • 1) and thair duala will f1\ltoJnCl.t1onllj' be nonbornorph10. When (12.11 does not hold we otumotaay that their w111
exist Ii. dual of a BIB doo:l.e.n 1... 0. a partial geometry (r,k,r), "110S8
etructure will exhibit the RG80oiat10n relati~ns~
In Berwral t!Hm wu can sny that it we ha.ve n l?liHJudo.. geo@Jnotxoio a880oi~.l t1,on Boheme wi th PHl'UI1l(' tore (~.,1). (4.2). then if
-
the
al.~l1ociation
~6
•
structure can be ex}}!11 ted by..,.p.0ans of U pHrtial
gE:lOCilEJtrs- (r,k,t), the first associates being thOaEltroHtUlEJnta which
oOrr{H31Wnd to points on a line of the gl'Jometr,y. Uuc;h flcllemeu muy
c·b.llad geollietrio aohelllea. 'lIhue when (12.2) ie true, the llS80ciution
Elchtlme will be (ietElndned up to tY'le, for there will exist nOll-isomor-
t.. 'l'hiB may be
regarded as a gim.(''l'ulized uni,!uenesG thuOrfJIll. Wlwu (12.2) is not
phic partial geometries with the
saSll(;)
pu-l'UI4J.etol·s r, k,
true very 11 ttlo is known except fOl' schemes which have the SfJUle
puramett.lra a.a the triangular schet'lc or the
Thuse
tWQ cas~e
L
2
scheme (r .. 2, k, t • 1:
have been tully investigated.
1'. A general embedding theorem.
'l'llOorcm
(13. 1). 01ven a. Pl3IIl deeigl1 (r,k,A ,A 2 ),
1
based on a s trongly
r~I~I.l.lnr
graph
A >A
2
1
G (aslJooiaT.ion soheme) with pt,ra-
meters
n1 .. (d-1)(k-1)(k-t)/t,
n
2
..
d(k-1),
P~1 .. [(d-1)(k-1)(k-t) - d(k-t-1) -~/t ,
P~1
• (d-1)(k-t)(k-t-1)/t.
we can extend the design by aflding new 't;looks, containing
t!.IO salTle
trulliJllents, in suoh a way that· the oxtended design is a balanced in-
complett: block (BIB) desie~n with r o • r + d(>'1 - >'2) replioations,
block size k and in which ev~ry pai~ ot troatments oocur together
in A1 blocks, provided that
Let G* be tho oomplolllentary of a, i.e. G* is the graph
with the same vertiouaaa G, but with the rrelution of adjaoenoy reversed, 1.0. Just thofJe vortices in G* aru Joined which were unjoined in G. This means that first auuoolatea becomo seoond associates and viae versa. Th~ parameters at a* aro obtained from G by
- "51 -
intorohanging the 8ubaorillta 0.11.11 8uporaoriptfJ
1
and
2. 11011.08 for
G*
(1'.3)
nf.
(1,.4) p~;
.
d(k-1),
nl • (d-1)(k-1)(k-t)/t,
[(d-1)(k-1)(k-t)-d(k-t-1)-t]/t,
P~;
.
(d-1)(k-t)(k-t-1)/t.
Using the identities (2.,), (2.4), we tind that
2*
P11 • d'\:.
i8 pseudo-goometrio wi til charaotoy'iotios (d,t, t).
In View o,r (1'.,) it tollows .from theorem (9.') that G*
is geometriot.\bls (d,k,t). From (4.1) tlle geometry hao lXx.c.jH'l.J(lf~Ij
Henoe
G*
d [(d-1 )(k-1)
+ t]/t blocks, and every pair at treatments whioh were
seoond Q19soo1o.toa in the original, paIn desian oocur once in tllfJ new
blocks. It we add these new blook. repeated A, - Aa times to the
original blooka then syoh pa.ir oocurs A, " times and eaoh treatment
oooura r + d(A , - A ) time•• This proves our theorem.
a
We ahall now drive trom this the embedding theorem on
orthogonal Latin squares due to Shr1khande and ~ruck.
In theorem (1,.1) take t .
then the 0* 18 a peeudog.ometric grapb (d,k,d-1) 1...... p.eudo-net sraph. It
i-',
it 1. geometr18able.
Alao let us take r . k + 1 .. d, A • 1. A • O. Then the
2
1
PBID dee1611 beoomes tbe design (k + 1 .. d, k, 1, 0) based on the
.trongly regular graph with p6rameters (7.1), (1.2). '1'h18 i8 easy to
oheak by IUbst1tutin« d., k +1 - r in (1,.1), (1,.2) and noting
that they reduoe to (7.1), (7,2). Henoe the PBIll design 18 a net ot
degre. k + 1 .. d, ar & lattioe design with r . k + 1 - d and blook
liz. k. Ilenae the extended design il a BID design with r + d i.e.
k + 1 replioations in whioh every pair ot treatments oooura in one
b1ook. This 18 an affine plane of order k. Henoe we have
Theorem (1,.2A). A lattioe de.ign (or a net) with
~ • k + 1 - d
and block aiz. k oan be oompleted to an affine plane
Ao • 1,
by adding kd new b1001s, it k>i (4-1)(4' - d2
+ 4+ 2).
the equivalenoe ot a lattioe
deeian with r rep1ioations and blook ei.e k, with a set ot r-2
autually' or'thogonal Latin squares ot order k. Since an arfine plane
ot order k oan be re,arded asa lattice with ,k+1 replioations,
theorem (1,.2A) may alternatively be otated as
1'heoreI4 (1,.2]3). It ther4IJ exi8t k-1-4 NU tually orthogonal
Latin aquurca at ordor k. it 18 possible to get a oomplote set of
k-1 mutull1y orthogonal LAtin squares, b1 adding d new Buitably
oho••n squares, provided that
(4..1)(4' - 42 + d + 2).
Tho oaa. d. 2, was first obtained by Shrikhande (20]
and the' ,eneral oaS8 was Obtained by Bruok [5].
Now we have already notifIed
k>t
- }9 -
Ite.ferenCOQ
(1)
(6)
(8)
It.C.
and
w.n.
Clatworth;t:: I "Some classes of partially
'balanced designs", Ann. Matb. stat., Vol. 26
(1955), pp. 212-2'2.
~.C. Doee and D.M. Meaner J
"On linear associative algebras
corrosponding to assooiation sohemes of partially
balanoed designs", Ann. Math. Stat., Vol. ,0
(1959), pp. 21·'0.
R.C. Boue and K.H. Uair I "Partially balanced lnooupleto blook
dCBiens", Sa.nkhya, Vol. 4 (19'9), PP. "1-312.
R.O. Bose and T. Shimamoto I "Claaeitioation and analya10 at
pllrtiallYbalanoed inoomplete block designs, with
two a60001u..te olaooes", J. Amer. Stat. Assn.,
Vol. 47 (1952), PP. 151-164.
B.H. Bruck' I II b~ini to nota II. Uniqueness and imbedding",
Paoifio J. Math.
Y.hang Li-ohien' .. fJ.'he uniquenes" and non-uniqueness of the
triangularaaaociation schemes", Science Heoord,
Math., New Ser., Vol. , (1959), pp. 604-61'.
Chong Li-ghiea I "Assooiation sohemes of partially balanced
d05igna with parameters v • 28, n , • 12, n2 • 15
and p~,. 4", Soienoe Hocord, Math., new Serf
Vol. 4 \1960), pp. 12-18.
'
W.ll. Clatworth;y I itA (jEloI!1otrioa.l configuration which ie a partially balanoed inoomplete blook design", Proe. Amer.
Math. poo., Vol. 5 (1954), pp. 41-55.
w.s. CO.rulO!: I, "Tbo uniq,uonoss of tho trin.ugula:r asoQcial,ion
achome ll , Ann. Math. stat., Vol. 29 (1958), pp. 262]3000
266.
(10)
(11)
(12)
(1 ,)
(14)
n.S. Connor,
"On the structure of balanoed incomplete blook
doa1gna", Atm. Math. Sta.t. Vol. 23 (1952),
pp. 57-71.
w. s. Connor and Vi .li. Clatwortb;t:: I "Somo theOrOIJ1U for partially
balanoed designs", Vol. 25 (1954), pp. 100-112.
Marshall Hall, and W.S. Connor I "An embedding theorem for
balanced inooluplete blook deu1gns", Can. J. Math.
Vol. 6 (1953), pp. '5-41.
A, J. lIoffrnnn I "On tho uniqueness of the tria.ngular nS8ooi('\. tion
soheme ll , Ann. Math. Stat. Vol. '1 (1960),
pp. 492-491.
A.J. Hoffman' "On tho exoeptional case in a ohn.raoteri~ation,
of the arcs of a oomplete graph", IBM J., Vol ...
(1960). pp. 481-496.
•
• 40 -
D.M. Meoner
"An investigation of cortain combinatorial pro-
I
portico of partially balancel incomplete block
experimental designs an~i asaocia tion schemes, with
a detailed study of (losigne of Latin equal's and
related types", unpublished doctoral theeis,
Michiaan state UniV81'uity, 1956.
(16)
E.J.li'. l'rimroae
(17)
D.l{. Ray.Chattdhuri
"Quadratios in ,lolin1te Guometriea", Frog. Cn:mb.
fhil. SOOt Vol. 41 (1951), pp. 299·'04.
I
I
jeotivG
( HI)
(19 )
(20)
(21)
"Some results on quadrics in finite proCan. J. Math.
"Applioation of the geometry ot quadrios
geome~r;y";
D.K. Ral-Chal\dhur.! I
tor oonstruoting £IBID designs '1 , Ann. Math. Stat.
.
' . .
tf
S.S. Shr1khande I "The uniqueness ot t}~ L2 as~ou~~tion.soh8m.
.Ann. Math. Stat. Vol. '0 (1959), PP' 781-790.
S.S. Shrikhanda, tfA nota on mutua.lly orthogono.l LfI.\,inaquares",
Sankhya, Vol. 23 (1961), pp~ 115-116.
.
8.S. Hhrik~anc!.8 a f'On a oharaoterization ot the triangular
, association soheme". Ann. Math. stat., Vol. '0
(1959),
(22)
PP. '9-47.
"Relations between oertain incomplete block
designs", ContributiOns to probr~b1lity and statioticB.Essays in honor of Harold' Motalling",
Stanford U. Press (1960), pp. ,UO-'95.
8.8. Shrikhande,
5.S. Shrikhande ,"On tho dual oreoms b1d.anoad incom:plete
block designs". B10motrlc:i8, Vol. e (1952),
~.rate8
I
pp. 66...72.
"Lattioe .quare,". J. Ag. So., Vol. 30 (1940),
pp.
672-687.
.