A CHARACTERIZATION OF THE GRAPH
OF THE.
. ASSOCIATION SCHEME
m
~
by·
Thomas A. Dowling
University of North Carolina
.Institute of Statistics Mimeo Series No. 535
June, 1967
Abstract
The graph G of the T association scheme is a graph on (fA)
. vertices, where 2 $ m $ n/2, suchmthat the vertices may be identified
with unordered m-plets (a l ,a , ••• a ) from among the n symbols 1,2, ••• >n
2
and two vertices of G are adJacentmif
and only if the corresponding mplets have exactly m-l common symbols. If d(x, y) denotes the distance
between two vertices x and y and .6o(x,y) denotes the number of vertices
adjacent to both x and y, then G is a connected graph possessing the
following properties: (a ) The number of vertices in G is (~); (a )
1
(a ) 6 (x,y) = n-2 if d(x,y) = 1; 2
G is regular of valence m\n-m);
(a4) 6 (x, y)~ = 4 if d(x,y) = 2. Irl this paper it is shovm that, conversely, if G is a connected graph and (a )-(a4 ) hold for some m > 2 and
n > 2m(m-l) ~ 4, then G is necessarily th~ graph of the T association
scheme. This result generalizes previously known resultsmfor the cases
m = 2, 3 corresponding to the triangular and tetrahedral schemes.
·i·
I
This research was supported by the National
: . ~cience Foundation Grant No. GP-3792 and the Air
-. ~Force Office of Scientific Research Grant No •
. !\F.-AFOSR-76o-65.
DEPARTMENT OF STATISTICS
UNIVERSITY OF NORTH CAROLINA
Chapel Hill, N.C.
"
.'.
I
II
A CHARACTERIZATION OF THE GRAPH
OF THE T ASSOCIATION SCHEME
In
by
I
Thomas A. Dowling
University of North Carolina
Chapel Hill, N. C.
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1.
Introduction.
An m-class association scheme, called the T scheme,
m
introduced in [7J as a generalization of the well-known triangular
scheme with t,vo associate classes. The T scheme may be defined as
vTaS
m
follows:
Let m and n be positive integers satisfying 2:: m:: n/2,
and suppose ,.,e are given v = (fa) treatments CPl,CP2' ••• 'CPv' To each of
the v unordered m-plets (al ,a , ••• ,am) of distinct symbols from among
2
1,2, ••• ,n, ,·re associate a treatment cp in some one-to-one manner. Two
treatments cp and cp' are said to be u-th associates of each other if and
only if the corresponding m-plets (al,~, ••• ,am) and (ai,a2, ••• ,a~)
contain exactly m-u symbols in common. The triangular association scheme
clearly corresponds to the case m = 2.
The parameters of the Tm scheme are given in [7J as follows: The
nUr.1ber of treatments is
(1)
v = (M.).
The number of u-th associates of each treatment is
(2)
n -_ (ID)(n-m)
u u,
u = O,l, ••• ,m.
u
For any two treatments cp and;j' vThich are t-th associates, the nwnber of
treatments which are s-th associates of cp and at the same time u-th
associates of cpT is
t =i~O
m:t (m-t)
( m-s-i
t ) ( m-u-i
t ) ( s+u-mf-i,
n-m-t. ) u,s,t = O,l, ... ,m,
Psu
i
,mere as usual we take (§) = 0 if x < y or y < O.
Given any association scheme, the £raph of the scheme is constructed
by identifying the v treatments with vertices, and joining two vertices by
an edge if and only if the corresponding treatments. are first associates.
This research was supported by the National Science Found.ation Grant
No. GP-3792 and the Air Force Office of Scientific Research Grant No.
AF-AFOSR-76o-65.
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Thus b'IO vertices of the graph G of a T scheme are adjacent if and only
m
if the corresponding m-plets have exactly m-l symbols in common. More
generally, one can easily ShOlf that two vertices of G are at distance u
from each other if and only if the corresponding m-plets have exactly m-u
symbols in common, i. e., if and only if the corresponding treatments are
u-th associates.
Using (1), (2), and (3) vIe have in particular for a Tm scheme v = Oft),
n l = m(n-m), pil = n-2, pil = 4. If for any finite, connected graph we
denote by d(x,y) the distance betvleen the vertices x and y, and by ~(x,y)
the number of vertices adjacent to both x and y, then the above parameters
imply that the graph G ofa Tm scheme possesses the following properties:
(al ) The number of vertices in G is (~);
(a ) G is regular of valence m(n-m);
2
(~)
~ (x,y) = n-2 if d(x,y) = 1;
(a4) ~ (x,y) = 4
if d(x,y) = 2.
In the case m = 2, corresponding to the triangular association schen~,
Connor [2J showed (using somewhat different terminology) that if n > 8
the triangular graph is characterized by properties (al )-(a4). Shrikhande
[8J, Li-Chien [5,6], and Hoffman [3,4J completed Connor's work by showing
that the same result holds if n < 8, but that if n = 8 then there exist
graphs satisfying (al )-(a4) i-Thich are not triangular. Recently Bose and
Laskar [lJ have investigated the graph of the T scheme, which they called
3
a tetrahedral graph, and were able to show that a connected graph satisfying (al )-(a4) for m = 3 is tetrahedral if n > 16. In this paper we
generalize these results to an arbitrary m ~ 2. The raain theorem states
that if G is a connected graph satisfying (al )-(a4) for some integers
m > 2 and n > 2m(m-l) + 4, then G is the graph of a T scheme. Equivalently,
m
we can sa:y that the T scheme is characterized by the four parameters
m
v, n l , Pil,pil when n > 2m(m-l) + 4. Little is knovffi at present concerning
the uniqueness of the T scheme when m > 3 and n < 2L1(m-l) + 4.
ill
- -
2. Characterization of the Tl'fl: graph.
We shall consider only finite,
undirected graphs G with no loops and no multiple edges. For completeness
it is convenient to recall here several definitions concerning such graphs.
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A chain C = (Xl' x2 ' •• " xt ) is a sequence of t :::- 2 d.istinct vertices of
G, not necessarily all distinct, such that any two consecutive vertices in
the sequence are adjacent. Tlms the pairs (Xl' x2 ), (X2 ' ~3), .. " (xt_l,X t)
are edges of G. The number of edges t-l is the length of C, and C is
saia. to join xl and xt '
G is a connected grapll if there exists a chain joining X and y for
every pair of distinct vertices x,y. For a connected graph the distance
d(x,y) is defined to be the length of the shortest chain joining X and y.
Clearly d(x,y) = 1 if and only if x and y are adjacent vertices. By
convention we take d(x,x) = 0.
The valence vex) of a vertex X is the number of vertices in G adjacent to x. If there exists an integer ~ such that vex) =
for every
vertex X in G, then G is said to be regular of valence n •
l
For any two distinct vertices x,y of G, we denote by ~(x,y) the
number of vertices adjacent to both x and y.Obviously ~(x,y) = if d{x,y) > 2.
If x and yare adjacent vertices, then ~(x,y) is called the edge-degree
of the edge (x, y). A regular graph G for which all edges have the same
edge-degree ~ is said to be edge-regular with edge-degree ~.
A clique K is a set of :mutually adjacent vertices in G. K is said
to be complete if there does not e:ld.st a vertex x ~ K such that x U K
is a clique.·
Vie nOvl quote a result due to Bose and Laskar [lJ \'lhich is of fundamental importance in the sequel. Consider an edge-regular graph G with
valence r{k-l) and edge-degree (k-2) + 0; such that 6. (x,y) ~ 1 + ~ for
every pair of non-adjacent vertices x,y, where r,k,o;,~ are fixed integers
nr
°
satisfying r :::- 1, k :::-- 2, 0; :::-- 0, ~ :::- 0, and r~-20; :::- O. A grand clique K in
G is defined to be a complete clique satisfying IK I : :- k - (r-l)O;. Then
we have the following theorem [lJ for G:
Theorem. If k > max [p{r,o;,~), p{r,o;,~)J, where
p(r,o;,~)
=
p{r,o;,~)
=
1
1 + 2 (r+l)(r~-~o;) ,
1 + ~ + (2r-l) 0; ,
then
(i) each vertex of G is contained in exactly r grand cliques;
iii
-
•
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(ii) each pair of adjacent vertices is contained in exactly one
grancl cIi que.
In the remainder of this paper vre shall denote by G a connected graph
p~ssessing properties (a )-(a4) of Section 1 for some integers m > 3 and
1
n > 2m(m-l) + 4.
Lemma 1.
Each vertex of G is contained in exactly m grand cliques,
and each pair of adjacent vertices is contained in exactly one grand
clique. If K is a grand clique, then IK I ~ n-m(m-l).
Proof.
If we set r • m, k = n-m+l, a = m-l, ~ = 3, then G satisfies
the conditions of the previous theorem. In this case we have
p (r a, ~) = 1 + . ~ (m+ 1)( m+2) ,
p(r,a,~)
=4 +
2
(2m-I) (m-l).
Thus
p(r,a,~) - p(r,a,~) = \~ (m-l) (m-2) > o.
Hence the theorem holds for G if .
n-m+l > 4 + (2m-l)(m-l),
i.e., if
n > 2m(m-l) + 4,
a condition which we have assumed to be satisfied.
By defi.nition we have for any grand clique K in G,
IK I ?
=
k .. (r-l)a
-m(m-l).
It is clear from LemJlla 1 that any two grand cliques in G can intersect
n
in at most one vertex. Also if x is any vertex and y is a vertex adjacent
to x, then y is contained in one and only one of the m grand cliques containing x. Since there are m(n-m) vertices adjacent to x by (~), and
each of these belongs to a unique grand clique through x, the average
number of vertices other tllan x in a grand clique containing x is n-m.
Thus the average clique size among the m grand cliques through x is n-m+L
The next two lemmas are needed to show that actually we have IKI= n-m+l
for every grand clique K in G.
Let K and K be t,.,o intersecting grand cliques in G, and let
o
1.
x = KO n Kl • Denote by E (KO' Kl ) the total number of edges (xo' xl) such
that X o E KO-x, X € K -x.
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LeIun.a 2.
For any t"ro intersecting grand cliques KO,K in G,
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E(KO,Kl ) 'S 1 + maxi IKo I, IKl l1.
Proof.
Let x = K n K • For each vertex X € KO-x d.enote by +(x ) the
O
l
o
o
number of vertices in Kl-x which are adjacent to xo. Also define
cp = . max
[ cp (x ) }.
o
X
E KO-x
o
We shall establish an upper bound for E(Ko,K ) for three cases correspondl
ing to different values of cpo.
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Let X € KO-x be such thatcp(xO) = C?
Denote the
o
cp vertices in Kl-x adjacent to X by Yl'Y2' ••• 'ycp. Since X \ K and K
o
o
l
1
is a grand clique and is therefore complete, there exists a vertex y € K
Ie
E(KO,Kl ) ~ 9.
Case 2. cp=2.
Again let X € KO-x be such that ':p (x )=2, and let
o
o
Yl 'Y2 be the two vertices in K1-x adjacent tox • If Y is one of the
o
IKll-3 vertices in K not adjacent to xo' then y can be adjacent to at
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most one vertex in KO-x, for otherwise we again contradict (a4). If
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Case 1. cp
~
3.
1
not adjacent to xo. Thus d(Xo'y) = 2, and by (a4) we have t::,. (xo,y) = 4.
But x,Y l 'Y2" •• 'Y are cp+l vertices adjacent to both X and Y, and so we
o
cannot have cp ~
Thus cp ~ 3 implies cp = 3. Again if 'P=3 and y is
adjacent to a vertex Z E KO-x, then xO,Yl ,Y 'Y3' z are five vertices·
2
adjacent to both X and Y, contradicting (a4). Hence the only vertices
o
in K1 -x which can be adjacent to vertices in Ko -x are Yl' Y2' Y3. Each of
these can be adjacent to at most three vertices in KO-x (including xo) -by
the arguY1l.ent above. Hence if 'iJ ~ 3 we have
4.
(4)
either Yl or Y2 is adjacent to three vertices in KO-x including xo' then
E(Ko,K ) ::: 9 by the argument in Case 1. Otherwise each of Yl and Y is
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2
adjacent to at most two vertices in KO-X including xO. Hence in any case,
if cp=2 then
(5)
E(KO,K ) '5
max
t 9, IKll+ 1 }
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Case 3. cp < 1.
In this case it is obvious that
(6)
E(KO,Kl ) 'S
/KO I - 1.
Now by Lemma 1 we have IKII ~ n-m(m-l). If IK11 + 1 < 9, then
n < m(m-l) + 8, which contradicts the assumptions m ~ 3 and n > 2m(m-l) + 4.
Hence combining (5) and (6) we have
E(KO,K ) <
max [ IK 1 1 + 1, IKo 1 - l}
l
1 + max l IKo I, I~ I } .
<
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Lemma 3.
If K is a grand clique in G, then IKI = n-m + 1.
Proof.
Let x be an arbitrary vertex in G and let KO,K , ••• , K _ be
mI
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the m grand cliques containing x. Let N = IKi I and assume without loss
i
of generality that NO ~ N1 -:: 0, •• ~ Nm_1 • Then by (a ). and Lemma 1 we have
2
~1 (N. - 1) = m(n-m)
~=o
~
or
-1
(7)
~O Ni = m(n-m+I).
Thus since the minimwn clique size can be no greater than the average, we
have in particular that
(8)
N < n-m+I.
o Let us now count the nTh~er of edges joining vertices of KO-x to
vertices of ~t (Ki-x). Corresponding to each vertex XOE: KO-x there are
No -2 vertices adjacent to both x and. X in Ko • By (~) 'I-re have 6(x'X )=n-2.
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He~ce there remain n-N vertices adjacent to both x and X which lie in
o
o
(~i-x). Since there are No-l vertices in Ko-x, the ~e~uired number
5r1edges is (n-No )(N0 -1). But this number is also given bY~ll
E(Ko,K.).
~~
Hence using Lemma 2 and (7) we have
}jl
(n-N )(No-l)
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i=l
E(KO,K.)
~
< ~l (N. + 1)
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= ~l
i~l
~
=
[m(n-m+l) - NO] + (m-l).
On simplification, this inequality becomes
~ - (n + 2) No
It follows that
(9)
where
(10)
I No
-
+ [(m+l)(n-m+l) + 2(m-l)]
> O.
1
~(n+2) I > ~
2
D,
D = (n+2)2 _ 4[ (m+l)(n-m+l) + 2(m-l)]
2
2
1
= n - 4mn + 4m - 8m + 8
= (n_2m-2)2 + 4(n-4m+l).
If n - 4m + 1 ~ 0, then since n > 2m(~-1) +4, we have
2m(m-l) + 4 < 4m - 1,
Le.,
2(m"1)(m-2) + 1 < 0,
"Thich of course is a contradiction. Hence n-4m+l > 0 and thus by (10)
'l'Te have
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1
(11)
2
D >n
2m - 2.
Then from (9) and (11) it follows that
-
- ~ (n+2) I > ~ (n-2m-2).
Suppose first that No < ~ (n+2) - ~ (n-2m-2) = m + 2. By Lemma 1,
2
NO :: n-m(m-l). Thus No < m+2 implies n < m + 2 and therefore
2m(m-l) + 4 < m2 + 2,
i.e.
. (m_l)2 + 1 < 0,
I NO
(12)
which is again a contradiction.
1
Hence by (12) we must have
1
No > 2 (n+2) + 2 (n-2m-2)
(13)
= n-.ffi.
From (8) and (13) and the fact that No must be an integer, it follows that
N = n-m+l. But NO < Nl < ••• < N 1· and ~l. N. = mN implies that
o
- mi::O. ~
O
NO = N = ••• = N = n-m+l. Hence each of the m grand cliques containing
l
m
x contains exactly n-m+l vertices. Since x is an arbitrary vertex and.
every grand clique is non-empty, it follows that IKI = n-m+l for every
grand clique K in G.
Corollary.
Pro:Jf.
For any tlVO intersecting grand cliques K ' K in G,
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E(KO' Kl ) '5 n-m+2.
This result follo't'rs immediately from Lemmas 2 and 3.
Lemma 4.
For any two intersecting grand cliques KO,K in G,
l
2
E(KO,Kl ) ? n-m +2.
Proof.
Let x = K n K and denote the remaining m-2 grand cliques conO
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taining x by K ,
,Km- l • Let us count the total nurnber of edges
2
connecting vertices of K. -x to vertices of K.-x
for all i, j = 0, 1, ... ,
~J
m-l, i:J:·j. If Y is a vertex in Ki -x, then there are n-m-l vertices in K
i
adjacent to both x and y. Since ~ (x,y) = n-2 by (a ), there are
3
(n-2) - (n-m-l) = m-l vertices adjacent to both x and y and not in K~.•
T11ese vertices must therefore liein~~ (Kj-X). By tw~ing each of the
m{n";m) possible choices for y, we ge~la total of m(m-l)(n-m) edges. But
each edge is included twice in this count, so that the total number of
distinct edges is ~ m(m-l)(n-m). Counting the number of such edges in
another obvious way, we obtain the relation
K3, ...
O'5i~-sm-l E (Ki , Kj )
= 21 m(m-l)(n-m).
Thus
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E(KO,K l )
m-l
rn.;:,l
m(m-.L)(n-m) - i~ E(KO,Ki ) -i~2 E(h·1,Ki )
1.
=2
r
-2~i<j5m-l E(Ki,K j
).
If we replace each of the E(· ,,),s on the right-hand side of (14) by
its upper bound n-m+2 as given by the Corollary, we obtain
E(KoK )
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Lemma 5.
>
1
1
2m(m-1)(n-m) - 2(m-2)(n-m+2) - 2(m-2)(m-3 )(n-m+2)
= n-m2 or 2.
If K is a grand clique and x is a vertex not in K, then there
are at most two vertices in K i·rhich are adjacent to x.
Proof.
Let t be the number of vertices in K adjacent to x and denote
these vertices by Yl'Y2""Yt'
The argument used in Case 1 of the proof
of Lemma 2 shows that t :: 4. . Suppose first that t=4 and d.enote by ~
the grand clique containing Yl and x.
Then since x is a vertex in KI-Y
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adjacent to three vertices Y2 , Y3~;Y4,in K-y , by Case 1 in the proof of
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Lemma 2 we have E(Kl,K) :: 9. Hence by Lemma 4 it follow's that n-m +2 :: 9.
Since n ~2m(m-l)+4, this implies
2
2m(m-l)+4 < m +
(m_l)2 < 4,
or
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which contradicts the assumption that m? 3.
Next suppose t
Hence we must have t < 3.
,IS
= 3.
Let K ,K
be the grand cliques containing x
l 2
Which intersect K in Y 'Y 'Y3' respectively. If Y is one of the n-m-2
l 2
vertices in K not adjacent to x, then d(x,y) = 2 and hence by (a4)'
6(X,y) = 4.
Thus there is exactly one vertex other than Yl'Y 'Y which
2 3
is ad.jacent to both x and y. Hence the number of ed.ges joining vertices
of K-Y -Y -Y to vertices of ia (Ki-x-y ) is at most n-m-2. It follows
l 2 3
l
i
that for at least one K , say K , the number of edges joining vertices
l
i
of K-Y l -Y -Y to vertices of Kl-x-y~is at most !(n-m-2). In addition
2 3
each of Y 'Y can be adjacent to at most two vertices in KI-Y including
2 3
l
x. Thus we have
(15)
E(K,~)
3(n-m-2) + 4 = !
(n-m+lO) •
2
2
By Lemma 4 (15) implies that! (n-m+10) ~ n_m +2, or 2n :: 3m -m+4.
Since n
..::
> 2m(m-l)+4, 2n > hm(m-l)+8 and hence we nmst have
2
4m(m-l) + 8 < 3m - m +
Le.,
(m-l)(m-2) + 2
<0 ,
8
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which again is a contradiction.
This proves the lemma.
Corollary. For any two intersecting grand cliques KO,K in G,
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E(KO,Kl ):: n-m.
~.
Let x = K n K1 • By Lemma 5 no vertex in KO-X can be adjacent
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to more than one vertex in K1-X. Thus the maximum number of edges is
at r.'lost n-m, the number of vertices in K -x.
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Lemma 6.
For any two intersecting grand cliques KO,K in G,
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E(KO' Kl ) = n-m,
and the vertices of KO-x (Where x = K n K ) may be put in one-one
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correspondence with the vertices of Kl - x so that corresponding vertices
are adjacent.
~.
To prove the first part we proceed exactly as in the proof of
Le,mma 4 except that we replace each of the E(' , ')'s on the right-hand
side of (14) by its upper bound n-m as given by the above Corollary.
We t11en obtain
E(KO,Kl )
? 21 m(m-l)(n-m)
1
- 2(m-2)(n-m) - 2(m-2)(m-3)(n-m)
= n-m,
which by the Corollary shows that E(KO,Kl ) = n-m.
If X o is an arbitrary vertex in KO-x, then by Lemma 5 there is at
most one vertex in Kl-x adjacent to xo. If there exists a vertex in
K -x 1n1ich is adjacent to no vertex in Kl-x, then we would have
E~KO,Kl) < n-m. Hence every vertex in KO - x is adjacent to exactly one
vertex in Kl-x. By the same argument, every vertex in Kl-x is adjacent
to exactly one vertex in K -x" and the lemma is proved.
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Lemma 7. The number of grand cliques in G is (m~l).
Proof.
Consider pairs (x, K) where x is a vertex in G and K is a grand
clique containing x. Since each vertex is contained in m grand cliques,
and by (al ) there are (~) vertices in G, the number of such pairs is m(~).
But each grand clique accounts for n-m+l such pairs, so the total number
of grand cliques in G is
m(~)/(n-m+l) = (mP-l).
Lemn~
8.
Each grand clique in G is intersected by exactly (m-l)(n-m+l)
other grand cliques.
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Proof. This follows at once by noting that each of the n-m+l vertices
in a grand clique is contained in exactly m-l other grand cliques.
Lemma 9. If K and K are ti'T8 intersecting grand cliques in G, then
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o
there are exactly n-2 grand cliques intersecting both K and K .
O
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Proof. Let x = KOn K • By Lerrnna 1 ther,e is a one-one correspondence
1
between edges joining vertices of KO-x to vertices of Kl-x and grand
cliques intersecting both K -x and K1-x. Hence by Lerrana 6 the number
O
of such grand cliques is n-m. In addition there are m-2 grand cliques
other than K and K which contain x. Thus the total number of grand
o
l
cliques intersecting both K and Kl is (n-m)+{m-2) = n-2.
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Lemrna 10.
If K and K are two non-intersecting grand cliques which
l
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are both intersected by a third grand clique L1' then there are exactly
four grand cliques which intersect both KO and ~, including L •
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Proof.
Let· Xo = K n L , xl = K n L • Since K and L are interO
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secting grand cliques, it follows from Lerrana 6 that there is exactly
one vertex Yl in Kl distinct from xl 'Which is adjacent to xO. Similarly
there is exactly one other vertex Yo in K adjacent to xl· Let L , ~.
O
2
be the grand cliques containing the pairs (xO'Yl)' (xl,yo), respectively.
Suppose YO and Yl are not adjacent. Then d(yO'Y l ) = 2. Since ~ and Kl
are intersecting grand cliques, there exists exactly one vertex zl in K
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distinct from xl and Yl which is adjacent to yO. Similarly there exists
a vertex Zo in K distinct from X and Yo which is adjacent to Y . But
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for the same reason there must also exist a vertex z2 in ~, distinct
from Xo and Yl , which is adjacent to Yl • Thus xo' Xl' zo' zl' z2 are
five distinct vertices adjacent to both YO and Y , which contradicts the
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fact that /::,. (Yo,Yl = 4 when d(yo'Yl ) =2 •. Hence ~t i'.Ollows that
YO and Y1 must be adjacent. Let L4 be the grand clique containing YO and Y •
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Suppose there is a fifth grand clique L Which intersects both
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KO and K]: Clearly L cannot contain any of the four vertices xO,xl'YO'Y l
5
or Lemr;m 5 would be contradicted. Let Zo = KO n L and zl = Kl n ~5'
5
There is a vertex z2 in Kl distinct from zl,xl'Y l which is adjacent to z00
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Zo and zl which is adjacent
xo' YO' zl' ;~2' z3 adjacent
There is also a vertex z3 in L distinct from
5
to xl' But then again we have five vertices
to both Zo and xl. Thus there cannot exist a fifth clique intersecting
both K and K and the lemma is proved.
O
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If G is the graph of a Tm association scheme, then G is connected
Theorem.
and the follovdng conditions are satisfied:
(a l ) The number of vertices in G is (~);
(a ) G is regular of valence m{n-m);
2
(~) 6{X,y) = n-2 if d{x,y) = 1;
(a4) 6{X,y) = 4 if d(x,y) = 2.
ConverSely, if G is a connected graph satisfying (a l )-{a4) for
some integers m ? 2 and n > ~m{m-l) + 4, then G is the graph of a Tm
association scheme.
Proof.
The necessity of the conditions (a )-(a4) follows from (I), (2),
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and (3) and can easily be verified directly.
The proof of sufficiency of these conditions When n > 2m{m-l) + 4
will be established by induction on m. Connor [2J has shmffi that the theorem holds in the triangular case, i.e.,when m = 2. Thus let m be an
arbitrary integer greater than two and assume that the theorem holds for
2,3, ••• ,m-l. Then since m? 3 and n > 2m(m-l) + 4, all the results
established above hold for G. Let H be a graph whose vertices are the
grand cliques of G, two vertices of H being adjacent if and only if the
corresponding grand cliques of G intersect. Since G is connected, it
is clear that H is also connected. By Lemmas 7,8,9 and 10 it follows that
H satisfies the following conditions:
(at) The number of vertices in H is (m~l) ;
1
(~)
H is regular of valence (m-l) [n-(m-l)J;
(~) 6(X,y) = n-2 if d{x,y) = 1.
(a4) 6(X,y) = 4
if d(x,y) = 2.
But these are just the conditions of the theorem with m-l replacing m.
Also n > 2m(m-l) + 4 implies that n > 2(m-l)[(m-I)" - lJ +4. Hence by
the inductive hypothesis the theorem holds for H, i.e., H is the graph
of a T 1 scheme. We can therefore associate the vertices of H with
m-
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(m-l) -plets from among the symbols 1,2, ••• , n. in such a vlay that two vertices
of IT are adjacent if and only if their corresponding (m-l)-plets contain
exactly m-2 common symbols. From the correspondence betvreen H and G it
follm'lS that 1'1"e can associate the grand cliques of G w"ith (m-l)-plets
from araong 1,2, ••• ,n so that two grand cliques intersect if and only if
their corresponding (m-l)-plets have exactly m-2 symbols in common.
Now let K and K be two intersecting grand cliques in G, and let
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(al ,a , .•• ,a _ ,a) and (al,a2, ••• ,am_2'~) be the corresponding (m-l)-plets.
2
m2
To the vertex x = KO n Kl we assign the m-plet (al,a2 ••• ,am_2,a,~). If
K is anyone of the other m-2 grand cliques containing x, then K intersects
both K and K , and therefore the (m-l)-plet corresponding to K must have
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m-2 symbols in common with both (al ,a2 , ••• ,am_2 ,a) and (al,a2, ••• ,am_2'~)'
Hence it must either be of the form (a l ,a2 , ... ,ai_l,ai +1,· •• ,am_2,a,~)
for some i, 1 ~ i ~ m-2, or else of the form (a ,a , ••• ,am_ ,r) where
2
l 2
Y
a, p. Suppose first that it is of the latter form. The total number of
(m-l)-plets which contain the (m-2)-plet (a ,a , ••• ,a _ ) is n-m+2, and
l 2
m2
corresponding to each of these is a grand clique in G. Now there are only
m grand cliques containing x and n-m+2 > m since n > 2m(m-l)+4, and thus
there exists a grand clique L not containing x whose corresponding (m-l)-plet
has the form (a l ,a , ••• a _ , l ), where ~~ aJ~,7. Then L is a grand clique
m2
2
which intersects Ko,K , and K. Thus x is a vertex not in L vlhich is
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adjacent to at least three vertices in L, which contradicts Lemma 5.
It fol.lovlS that if K is any grand clique containing x other than K and K ,
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then the (m-l)-plet corresponding to K must be of the form (al,~, ••• ,ai_l'
a.L1, ••• ,a 2' a~) Hence the m-plet (al ,a , •••A ,a,~) associated with
2
~'"F
m-"
m-2
x is unambiguously determined by any two of the m grand cliques intersecting
in x, and to each of the (~) vertices in G we can associate a unique m~plet.
If two vertices x and y of G are adjacent, then there is a unique
grand clique K containing x and y. If (a ,a , •.• ,a _ ) is the (m-l)-plet
l 2
ml
corresponding tJ K, then the m-plets corresponding to x and y must contain
the symbols a ,a , .•• ,a _ • Conversely, if the m~plets corresponding to
l 2
ml
x and y contain the symbols a ,a-, ... ,a l' then x and y are contained in
1 0::::
mthe grand clique K corresponding to the (m-l)-plet (al,a , ••• ,a _1 ). Thus
m
2
two vertices of G are adjacent if and only if the corresponding m-plets
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have m-l symbols in common.
Hence G must be the graph of a T
m
scheme.
AclmovTledgements
MY thanks are offered to Professor R. C. Bose and Dr. Renu Laskar
for the problem considered which is implicitly suggested by their paper
[lJ on tetrahedral graphs.
I I m also very grateful for the time and effort
they devoted' to discus'sing the problem with me.
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References
Bose, R. C. and Renu Laskar, A characterization of tetrahedral
graphs, lnst. Stat., UNC, Himeo Ser. No
in
[2J
l.
~.
Connor, W. S,
The uniqueness of the triangular association scheme,
29 (1958), 262-266.
Hoffman, A. J~, On the uniqueness of the triangular association
scheme, ~. ~. ~., 31 (1960), 492-497.
[4J
Hoffman, A. J., On the exceptional case in a characterization of
IBM J., 4 (1960), 487-11-96.
the arcs of a complete graph,
[5J
Li-Chien, Chang,
The uniqueness and non-uniqueness of the
triangular association schemes, Science Record,
~. ~~.,
3 (1959), 604-613.
[6J
Li-Chien, Chang, Association schemes of partially balanced designs
"lith parameters v
= 28,
lli:!!
~.,
Record., ~.
[7J
Ogasawara, M.,
nl
= 12,
n
2
= 15,
and pil
= L~,
Science
4 (1960), 12-18.
A necessary condition for the existence of regular
and synnnetrical PBIB designs of T type, lnst. Stat., UNC, Mimeom
Ser. No. 418.
[ 8J
Shrilmande, S. S.,
On a characterization of the triangular
association scheme, ~. Hath. ~., 30 (1959), 39-lt7.
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To be published
Theory.
~. ~. ~.,
[3J
509.
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