Tilburg University

Association schemes

Haemers, W.H.; Brouwer, A.E.

Publication date:1987

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Citation for published version (APA):Haemers, W. H., & Brouwer, A. E. (1987). Association schemes. (Research memorandum / Tilburg University,Department of Economics; Vol. FEW 258). Unknown Publisher.

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ASSOCIATION SCHII~S

W.H. HaemersA.E. Brouwer

~W 258

ASSOCIATION SCHEMES

A.E. Brouwer,Technological University Eindhoven

W.H. Haemers,Tilburg University

This paper gives a brief survey of the theory of Association

Schemes. It is destinated to become a chapter of the Handbook

of Combinatorics.

Table of contents

1. introduction 1

2. Strongly Regular Graphs 2

3. Association Schemes 4

4. Applications S

Association schemes.

A.E. Brouwer á W.H. Haemers

1. Introduction.Association schemes are by faz the most important unifying concept in algebraic combinatorics.

They provide a common view point for the treatment of problems in several fields, such as coding theory,design theory, algebraic gaph theory and finite goup tl~eory.

Roughly, an association scheme is a very regular partition of the edge-set of a complete graph Eachof the partition classes defutes a graph, and tbe adjacency matrices of these graphs (together with the iden-

tity matrix) form the basis of an algebra (known as the Bose-Mesner algebra of the scheme). But since this

basis consists of 0-1 matrices, we see that the algebra is not only closed under matrix multiplication, butalso under componentwise (Hadamard, Schur) multiplication. The interplay between these two algebra

structures on the Bose-Mesner algebra yields strong information on the structure and parameters of an

association scheme.

The relation of the theory of association schemes to the various fields varies. Finite group theorymainly serves as a source of examples - any generously transitive permutation representation of a goupyields an association scheme -, but also as a source of inspirauon - many goup-theoretic concepts andresults can be mimicked in the (more general) setting of association schemes. On the other hand, in com-binatorics association schemes form a tool. Using inequalities for the par~ameters of an association scheme,one finds upper bounds for the size of error-correcting codes, and lower bounds for the number of blocksof a t-design. Many uniqueness proofs for combinatorial structtaes depend crucially on the additional infor-mation one gets in case such inequalities óold with equality.

The graphs of the partition classes of an association scheme are very special. For irutance distance-regular graphs and ICneser graphs are of this type. For these graphs the spectrum of the adjacency matrixcan be computed from the parameters of the association scheme. Thus result.c about the eigenvalues ofgraphs (cf. Godsil's chapter) can be applied.

An association scheme with just two classes is the same as a pair of complementary strongly regulaz

gaphs. Because strong(y regular gaphs are themselves important combinatorial objects and because their

treatment fonns a good introduction to the more general theory, we shall start with a section on strongly

regular graphs. In the subsequent section we treat the elements of the general theory of association

schemes: the adjacency matrices, the Bose-Mesner algebra, the eigenvalues, tt~e ICrein pazameters, the

absolute bourxi and Delsarte's inequality on subsets of an association scheme. The Last section is devoted

to some special association schemes and the significance to other fields of combinatorics, as indicated

above.

Association schemes were introduced by Boss 8z Stm~tnMOro [lOJ and Boss 8c MESrr~e [9] definedthe algebra that bears their name. DstsatTE [17] found the lineaz programming bound, studied P- and Q-polynomial schemes, and applied t17e resulting theory to codes and desigru. I~oMwrr [29] inoroduced themore general concept of coherent configuration to study permutation representations of finite goups.There is much literature on association schemes and related topics. Bxrrrrnt ~ Iro [3] is the fust text bookon association schemes. On the more special topic ofdistance-regular graphs a lot ofmaterial can be foundin Bicas [7]: a monograph on this topic by Brouwer, Cohen 8c Neumaier is under preparation. Some intro-ductory papers on association schemes are DFaswie~tf: [I8], Gortrtws [23], chaptcr 2l of MwcWru.unvs 8cSLOAYE [~3 ], HnE.~.uxs [25 J, chapter 17 of Cwt~ROV 8c Vnr I~rr [ t S[:utd SEroa. [50].

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2. Strongly regular graphs.A simple graph of order v is strongJy regular with parameters v, k, ~., {t whenever it is not complete

or empty and(i) each vertez is adjacent to k vertices,

(ii) for each pair of adjacent vertices there are k. vertices adjacent to both,

(rii) for each pa'v of non-adjacent vertices there are lt vertices adjacent to both.

For example, the pentagon is strongly regular with parameters (v,k, ~.,{t) -(5,2,0,1). One easily ver-

ifies that a graph G is strongly regular with parameters (v,k, ~„it) if and only if its complement G is strongly

regular with parameters (v,v-k-l,v-2ktlt-2,v-2kt1t.). The line graph of the complete graph of order rn,

known as the triangular graplt T(m), is strongly tegular with parameters (Zm(m-1), 2(m-2), m-2, 4).

The complement of T(5) has parameters (10,3,0, t). This is the Petersen graph.

A gr,tph G satisfying condition (i) is called k-regulaz. [t is well-known and easily seen that the adja-cency matrix of a k-regulaz graph has an eigenvalue k with eigenvector j(the all-one vector), and that

every otlkr eigenvalue p satisfies ~p~ 5 k(see Godsil's chapter or Btoos [7]). For convenience we call an

eigenvalue resrrirtedif it has an eigenvector perpendiculaz to j. We let I and J denae the idenáty and all-

one matrices, respecávely.

2.1. Theorem. Fora simple graph G of order v, not complete or empty, with adjacency matrix A, the fol-

lowing are equivalent:

(i) G is strongly regularwith parameters (v,k, ~„lt) for certain integers k, À,, lt.

(ii) AZ -(Á-lt)A t(k -(t)I tlt.1 for certain reals k, ~„ lt.

(iii) A hasprecisely two distinct restricted eigenvatues.

Proof. The equation in (ii) can be rewritten as

A'--klt ~A tlt(J -I-A).

Now (i) ~(ii) is obvious. (iij ~(iii): L.et p be a restricted eigenvalue, atxl u a corresporxíing eigenvector

perpendicular to j. Then Ju - 0. Multiplying the equation in (u) on the right by u yields

p' -(1,.-lt)pt(k -lt). This quadraác equaáon in p has two disánct soluáons. (Indeed, (1l-lt)Z - 4(lt-k)

is impossible since lt 5 k and k, S k-1.)(ui) ~(ii): I.et r and s be the restricted eigenvalues. Then (A -rlxA -sl) - a! for some real number a.

So A 2 is a lineaz combinaáon of A, I andJ. ~

As an application, we show that quasisymmetric block designs give rise to strongly regular graphs. Aquasisymmetric design is a 2-(v,k, k.) design sucó that any two blocks meet in either x or y points, for cer-tain fixed.r, y. (Cf. the chapter on block designs.) Given this situaàon, we may define a graph G on the setof blocks, and call two blocks adjacent when they meet in x points. Then there exist ccefficientsa~ ,.-., a~ such that NN' - a~ 1 t a2J, NJ - a3J, JN - asJ, A - aSN'N t otbl ta~J, when; A is the adja-cency matriz of the graph G. (The ot; can be readily eapressed in terms of v, k, k., x, y.) Then G is stronglyregular by (ii) of the previous theorem. (Indeed, from the equaáons just given it follows straightforwardlythat AZ can be expressed as a lineaz combinaáon of A, 1 and J.) A large dass of quasisymmetric blockdesigns is provided by the 2-(v,k,7~) designs with ?` - I(also known as Stcirtcr systcros S(2,k,v)) - suchdesigns have only two intetsecáon numbers since no two blocks can meet in more than one point. Tttisleads to a substanáal family of svongly regular graphs, including the triangulat graphs T(m) (derived fromthe trivial design consisáng of all pairs out of an m-set).

Another concection between strongly regulaz graphs and desigrts is found as follows: Let A be theadjacency matriz of a strongly regular graph with parameters (v,k, k.,k,) (i.e., with ~. - p: such a graph issometimes called a(v,k, k.) graph). Then, by (2. l.ii)

AA' - A Z -(k -~.)1 t a.l.

whicó retlects that A is the incidence matrix of a squar~e (`symmetric') 2-(v,k,k.) design. (Atd in this wayone obtains precisely all square 2-designs possessing a polarity without absolute poirNS.) For instance, thetriangular gr;tph T(6) provides a square 2-(15,8,4) design, the complementary design of the design of

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poínts and planes in the projecáve space PG (3,2). Svnilarly, if A is the adjaceacy matrix of a strongly reg-ular graph with parametets ( v,k, k.,k.t2), then A t! is the incidence matrix of a square 2-(v,k,1) design (andin this way one obtains pcecisely all square 2-0esigns possessing a polarity with all points absolute).

2Z. Theorem. Lrt G be a srrongly regular grapl: wirh adjacenty matrix A and parameters (v, k, À. ,11). Let r

and s ( r ~ s) be the restricted eigenvalues ofA and let f, g be rheir respectivt multiplictties. Then

(i) k(k-1-A)-1t(v-k-1),(ii) rs-)t-k, rts-J~-lt,

~ (rtsxv-1)t2k(w) f~8- 2(v-13: ).

r-s

Proof. ( i) Fiz a vertex x of G. Let I'(x) and 0(x) be the sets of vertices adjacent and noo-adjacent to x,

respecávely. Counting in two ways the ntunber of edges between I'(x) and A(x) yields (i). ~tte equaáoas

(ii) are direct consequences of (2.1.u), as we saw in the proof. Formula ( iii) follows from ftg - v-1 andO-trA-ktfrtgs-kt 2(rtsxf'tg)t Z(r-sxf'-g). ~

These relaáons imply restricáons for the possible values of the pazameters. Clearly, the right hartd

sides of ( iii) must be positive integers. ~tese are the so-called ratronality condiáons. They imply that r

and s must be integers if f f g. As an example of the applicaáon of the raáonality condiáons we can derive

the followíng result due to HoHr-ntnrr 8t Suacts~roN [32].

2.3. Theorem. Suppose ( v,k, 0,1) is the parnmeter set of a strongly regular graph. Then ( v,k) -(5.2),

(10,3), (50,7) or (3250,57).

Proof. The raáonality conditions imply that either f- g, which leads to (v,k) -(5,2), or r-s is an integer

dividing (r ts)(v - L) t 2k. By use of (2.2.i-ii) we have

s --r-l, k -r2trtl, v -r"t2r;t3rZt2rt2,

ard thus we obtain r- 1, 2 or 7. ~

The first three possibiliáes are uniquely reafi~ed by the pentagon, the Petersen graph and the Hoffman-

Singleton graph. For the last case existence is unknown ( but see Asctmnct~tt [ 1]).

Except for the raáonality condiáons, other restrictions on the parameters are known. We mention

three of them.7he Kmin condiáons, due to Scarr [48], can be stated as follows:

(r t 1)(k tr t 2rs) 5(k t rxs t 1)Z,

(stlxktst2rs)5(ktsxrtl)Z.

Seidel's absolute bound ( see DEt.snx~ts, GoErtw.s 1~ St:.rotz. [20], KootsxwQrt~x [37J, Se.rn~[. [49)) reads

v5f(ft3),

v~8(Rt3).

The conference matrix condiáon, due to Btl.cvrrcx [4] (see also V~ Lmt 8c St~L ( 41p, states that if

f- g, then v must be the sum of two squares ( such a graph is related to a conference matrix of otder v t l).

The Krein conditions and the absolute bound are special cases of gener.tl inequaliáes for associaáon

schemes - we'll meet them again in the cezt secáon; the conference matria cortdition is the analogue of the

Bruck-Chowla-Ryser theorem for square 2-desígns. In BROtrw~ 8c Vnx Ltrrr [1 l] one may find a list of all

known restricáoa5; this paper gives a survey of the recent results on stmngly regulaz graphs. It is a sequel

to Husn~-r [33]'s earlier survey of constructions. St~- [49] gives a good tieatment of the theory. Some

other references are Bose [8], C.~.xoN [l4] and Cw~.teorr 8c V,uv Lu.rr [IS].

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3. Association schemes.

An association scheme with dclasses is a finite set X together with dtl relations R; on X such that

(i) ( Ro,R r, -.., Ra ) is a partition ofXxX;

(ii) Ro - [(x,X)~x E X};

(ui) if (x,y) E R; , then also (y.x) e R;, for all x.y E X and i E { 0, .~., d};

(iv) for any (x,y) e Rk the numberpj of z e X with (x,z) E R; and (z,y) E R~ depends only on i, j and k.

The numbers p~ are called the inrersecrinn numbers of the association scheme. The above definition is the

original definition of Bosa Bc Srm.tnt.toTO [l0]; it is what D~.snu~ [l7] calls a symmetric association

scheme. In Delsarte's more general defutition, ( iii) is replaced by:

(uí ) for each i E { 0, ~.-. d} there exists a j E{ 0, ~.~ , d} such that (x, y) E R; implies (y,x) E R~,

(~~)P~-P~.foralli,j,kE{0,.~ ,d}. ~

D.G. HicM:w [29, 30, 3l] studied the even more geceral concept of a coherent configuration. He nxluires

(i), (iii ), ( iv) and

(ri~ ((x,z)~x E X} is a union of some R~.

If (ii) holds, then the coherent configuration is called homogeneous. We shall not treat coherent configura-tions here, but content oulselves with the n;madc that a coherent configuration with at most 5 classes mustbe an association scheme ( in the sense of Delsane), see HicntwN [30].

Define n- ~X~, and n; :-p,~. Clearly, for each i E (1, ~.., d}, (X,R;) is a simple graph which is reg-

ular of degree n;.

3.1. Theorem. The intersection numbers of an association scheme satisfy

k 0 k k(i) Po~ -s~k, p;~ -s~~n~~ P~i -P~;.

(li) ~Pj - ni. ~ ni - n., ~

(iii) p~nk - p~kni.

(iv) r , r m~P:~Pir - ~Pk~P:r .~ r

Proof. (i)-(iiij are straigfrtforward. The eapressions at txxh sides of (iv) count yuadruples (w,.r,y,z) with

(w,x) E R;, (x.y) E R~, (y,z) E Rk, for a 5xed pair (w,: ) E R,~. [i

It is convenient to write the intersection numbers as entries of the so-called inrersecrion marrices Lo, -- ,

Ld:k

(L~)kj -P;~.

Note that Lo -1. From the deónition it is cleaz that an association scheme witó two classes is the same as apair of complementary strongly regulaz graphs. If (X,R r) is strongly regulaz with parameters (v,k, ~.,it),then the intersection matrices of the scheme are

0 k 0 0 0 v-k-IL~- l k. k-k,-1 LZ - 0 k-~.-1 v-2k t~,

0{t k-)t l k-lt v-2 ktlt-2

Wr ,cc Ihat (iii) gcncralius (2.2.i).

The Itose-Mesner algebra.

The mlations R; of an association scheme are described by their adjacency matrices A; of oreler n

drfined by

(Ar)ry -~ l whenever (.r,Y) E R;,

I 0 otherwise.

In other words, A; is the adjacency matriz of the graph (X,R;). In terms of the adjacency matrices, the

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azioms ( i}(iv) becomed

(i) ~A; -J.;~

(u) Ao -~,(iii) A; -A', for all i e[ 0, -.., d},

(iv) A,Ai-~p~At,foralli,j,ke [0,"',d].k

From (i) we see that the A; are linearly independent, and by use of (ii)-(iv) we see that they geoerate a cmn-mutaàve (dfl)-dimensional algebra A of symmetric matrices with constant diagonal. This algebra wasfirst studied by BosE 8c MFSr~x [9] and is called the Bose-Mesner algebra of the associaàon scheme.

Since the matrices A; commute, they can be diagonaliud simultaneously (see Mwttcus 8c Mare [44]),

that is, there exist a matrix S such that for each A E A, S-~AS is a diagonal matriz. Therefore A is semisim-

ple and has a unique basis of mínimal idempotents Eo, ..-, E„ (see Buxxow [12]). These are matrices

saàsfying

E;E~ - S,~E;,

d~E; - I.~-o

The matrix ~J is a minimal idempotent ( idempotent is cleaz, and minimal follows since rk! - 1). We shall

take Eo - ~J. L.et P and ~Q be the matrices relaàng our two bases for A:

dA~ - ~,P;~Er,

r~d

EI - n~Q(jA;.i~

'Ilten clearly

PQ - QP - nI.

R also follows that

A~E; - P;~E;,

which shows that the P;~ are the eigenvalues of A~ arxi that the columns of E; ane the corresponding eigen-

vectors. T'hus lt; : rk E; is the mulàplicity of the eigenvalue P;~ of Ai (prnvided that P;i tPk~ for k~ i).d

We see that ~- l, ~lt; - n, and )t; - tr E; - n(E;)ii (indeed, E; has only eigenvalues 0 and 1, so rkEk~~

equals the sum of the eigenvalues).

3.2. Theorem. The nurnbers P;~ and Q;~ salisfy

(i) Pro-Q;o-1.Po;-n;,Qa-Fir~d

f(u) P;~P;k - ~PikPrn

r~

(~) Ft;P;~ - n~Qir, ~Ft~P;iP;k - nn~sik, ~nrQ;~Q;r - nk~sik~

(iv) ~Pr;~ sni~ ~Qr~~ ~Fti.

Proof. Pan (i) follows easily from ~E; - I -Ao, ~A; -J- nEo, A;J - n;l, and tr E; -~t;.

Pari (ii) follows from A~A4 - ~PjkAl.I

The tirst equality in (iii) follows from ~ n~Q~;P;k - nn~Sik - tr A~Ak -~{t;P;~P;k, since P ís nonsingular,

this also proves the second equality, and the last one follows since PQ - nI.lite first inequaliry of (iv) holds because the P;~ are eigenvalues of the n~-regulaz graphs (X,R~). The

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second inequality then follows by use of ( iii). ~Relations ( iii) are often teferred to as the orthogonaliry relations, since they state that the rows (and

columns) of P (attd Q) are orthogonal with respect to a suitable weight function.

If d - 2, and (X,R ~) is strongly tegular with parameters (v,k, ~.,It), the matrices P and Q are

P- . Q-

I Í RI r s

fk 8kI rtl - stl

~ -f v-k-1 -R v-k-l

where r, s, f and g can be expresxd in terms of v, k, k by use of (2.2).

In gener:tl thc matrices P and Q can be computed from the intetsection numbers of the scheme, ysfollows from the following

3.3. Theorem. For i - 0, .-., d, the intersection matrix L~ has eigenvalues P;i (0 5 i 5 d).

Proof. (3.2.ii) yields

P~i~P;k(P-t)~ - ~,Prr(Li)!k(P-1)~.k k,!

hence PLiP-~ - diag (Poi, ~ ~ ~ , Pdi) ~Tt~anks to this theorem, it is relatively easy to compute P, Q(- nP'~ ) and ~t; (- Qo;). It is also

possible to ezpress P artd Q in terms of the (common) eigenvectors of the L~. Indeed,PLjP-~ - diag (Poi, ~.-, Pai) implies that the rows of P are left eigenvectots and the columns of Q areright eigenvectors. In particular, lt; can be computed from the right eígenvector u; and the left eigenvectorv,', nomt:dized such that (u;)n -(v;~ - 1, by use of lt;u'v; -n. Clearty, each (t; must be an integer. Theseare the rationaliry conditions for an association scheme. As we saw in the case of a strongly cegular graph,these conditions can be very powerful. Godsil ( see his chapter) puts the rationality conditions in a moregeneral fonn, which is not restricted to association schemes.

The Krein parameters.T'he Bose-Mesner algebra A is not only closed under ordinary matrix multiplication, but also under

componentwise (Hadamard, Schur) multiplication (denoted o). Clearly [Ao, ~~., A~ [ is the basis ofminimal idempotents with respect to this multiplicatioa

Writee

E;oEi- ~ ~4~Ek.~ k -0

The numbers qj thus defined are called the Krein parameters. (Our qj are those of Delsarte, but differfrom SErotl. [49]'s by a factor n.) As expected, we now have the analogue of (3. t) and (3.2).

3.4. Theorem. The Krein parameters ofan association scl~eme satisfy

(i) qOj - sj4. qo - óri{~i. qj- qjr.k

(u) ~9;j-Fti~ ~pi-n.; j

(ui) 4 ~Ílk - 4]k4rj,

(tv) ~q~j4kl-~4kjq;r.

(V)

r rcd

Qi~Qik - (~qikQil.I~0

(V!) nitlq j-~nlQLQfjQlk~(

Proof. L.et E(A ) denote the sum of all entries of the matriz A. Then JAJ - E(A ~I, E(A oB )- tr AB' ancí

-~-

F.(E,)- 0 if i~ 0, since then E;J- nE;Eo - 0. Now (i) follows by use of E; oEo -~E; ,

q~- F.(E;oEi) - tr E;Ei - S;;Iti, and E; oEi - E~oE;, respecávely. Equaáon (iv) follows by evaluatingE, oE~ oEt in two ways, and (iii) follows fiom (iv) by taking m- 0. Equaáon (v) follows from evaluaángA; oE~ oEt in two ways, and (vi) follows from (v), using the orthogonality relaáon ~ ntQ~mQtt - S,,,kWkn.

rFinally, by use of (iii) we have

4tk~4j -~4]kFti - n.tr (E; oEk) - n~(E~)t~(Ek)u - Lt~ltk~

proving (ii). ~

The above results illustrate a dual behaviour between ordinary multiplicaáon, the numbers p~ andthe matrices A; and P on the one hand, and Schur multiplicaáon, the numbets q j and the mattices E; and Qon the other hand. If two associaáon schemes have the property thaz the intersection numbers of one arethe Krein parameters of the other, then the converse is also true. Two such schemes are said to be (for-mally) dual to each other. One scheme may have several ( fonnal) duals, or none at all (but when thescheme is invariant under a regular abelian group, there is a natural way to define a dual scheme, cf.DasaelE [I7]). In fact usually the Krein parameters are not even integers. But they cannot be negaáve.These important restricáons, due to Sco~rr [48], are the so-called Krein conditions.

3.5. Theorem. The Krein parameters of an association scheme satisfy qj? 0 for al! i, j, k e ~ 0, .~-, d[.

Proof. The numbers '-q;; (0 5 k 5 d) are the eigenvalues of E;oEi ( since (E;oEi)Ek - nq~Et). On the

other hand, the Kronecker product E; ~ E~ is posiáve semidefinite, since each E; is. But E; oE~ is a princi-pal submatrix of E; ~ Ei, and therefore is posiáve scmidefinite as well, i.e., has no negaáve eigenvalue. ~

The Krein parameters can be computed by use of eyuaáon (3.4.vi). This equation also shows that theKrein condition is eyuivalent to

~ n~Q;,Q;~Q;k ? 0 Cor all !, j, k E(0, ..., d ~.

In case of a strong)y regular graph we obtairt

~ r3 (rtl)3 ~0,9it-~ It--v k2 (v-k-1)2

Z ~(l} s3 (stl)3 1 ~~qzz - v Il k'- - (v-k-1)ZJ

(the other Krein conditions are trivially satisfied in this case), which is equivalent to the result menáoned inthe previous secáon.

Nsuntw~R [45], generalized Seidel's absolute bound to association schemes, and obtained

3.tí. Theorem. The multiplicities {t; (0 5 i 5 d) of an association scáeme with d classes sadsfy

~ pk ~y'y:o

It;lt~ if i ~ j,

~)1;(It; t l) íf i- j.

Proof. The left hand side equals rk (E;oEi). But rk (E, oEi) 5 r1c (E; ~ Ei) - rk E; x rk Ei - It; lti. And ifi- j, then ric(E;oE,) 5 Z-{t;()t; t 1). Indeed, if the rows of E; are linear combinations of p; rows, then the

rows of E; oE; are linear combinaáons of the It; t~)t;(W; - 1) rows that are the elementwise products ofany

two of these )t; rows. ti -

For strongly regular graphs with q i~- 0 we obtain Seidel's bound: v 5 2f(jt 3 ). But in caseq~i ~ 0, Neumaier's result states that the bound can be improved to v 5 Zf(jt 1).

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Subsets of association schemes.The last subject of this secáoo is a result of D~snn~ [l7] (Theorem 3.3, p. 26) on subsets in associ-

aáon schemes. For Y~ X, Y x 0, we define

a; - iYl x'A;x,i

where X is the characterisàc vector of Y. In other words, a; is the average degree of the subgrapó of (X,R; )d

i~uced by Y. Clearly ao - 1, and ~a; - ~Y~. The vector a-(ao, ~~~, ad) is called the inner distribution;~

of Y.3.7. Theorem. The inner distriburion a of an nonempry subset of an association scherne satisfies aQ 2 0.

d dProof. ~Y~ ~a;Q;i-~zTQ;iA;x - nxTEix ? 0, since Ei is posiàve semidefinite. ~

i~ ;~This inequality leads to Delsarte's Linear programming bound, as we shall see in the cezt secàon.As an application we have the following tesult (Dtasat~ [ 17], Theorem 3.9, p. 32).

3.8. Theorem. Let (( 0 ~, I~, !Z ~ 6e o partition of ( 0, ~~-, d(, and assume thar Y and Z are nonempry sub-sets ojX such that the inner distribution b of Y sarisfies b; - 0 for i E 1~, and the inner distribution c of Zsatisfies c; -0 for i e IZ. Then ~Y~ ~Z~ 5 ~X~, and equaliry ho[ds if and only if for all i x0 we have

(bQ)i - 0 ar (cQ)~ - 0.

ProoP. Define ~i; -1t;~~Z~-~~c~Qi;. Then ~-1, p;?0 for all i, and ~p;Qk;-nk~~Z~-~ncR. Now wei ~

have ~Y~ -~64 -(bQ)o ~ ~bkQk;~r -~Qorai - iZi . ~k i,t ,

Let us inveságate a special case of this situaàon somewhat closec Let, for I~ ( 0, -.-, d[, the !-spherearound the point .r e X be the set ( y e X ~(z, y) e R; for some i e! ~. A nocempty subset Y of X is calledperfect (more precisely, !-perfeet) when the !-spheres around its points form a partition of X.

3.9. Theorem. ('Lloyd's theorem', cf. Ltovo [42], Det.snx~ [17], p. 63}. Ler Y be !-perfect, with innerdistribution a. Then ~ P~; - 0 for all j x 0 such that (aQ)i x 0.

iEl

Proof. Apply the previous theorem, with for Z an !-sphere. Ifc is the inner distribuáoa of Z, then

~Z ~lt~ ~ (cQ )i - Wj 1 ~ ~ nxPanQri - ~ ~ PakP~i - ~ PiRPiti - ( ~Pii)2. ~i,q,hEl i R.hEt g.hEt iEI

4. Applications.

In this secàon we discuss some special types of associaáon scheme and their significance to other6elds of combinatorics.

Distance regular graphs.

Consider a connected simple graph with veRez set X of diameter d. Define R; eX2 by (x,y)e R;whenever .r and y have graph distance i. If this defines an associaàon scheme, then the graph ( X,R ~) iscalled distance-regular. Tbe cortesponding associaàon scheme is called rnetric. By the triangle inequality,pk~ - 0 if i t j ~ k or ~i - j I ~ k. Moreover, p; ji ~ 0. Conversely, if the intersecáon numbets of an associa-tion scheme saásfy these condiàons, then (X,R ~) is easily seen to be distance-regular.

Many of the associaáon schemes that play a rdle in combinatorics are metric. In fact, all the ezam-ples trcaled in this chapter :ur metric. Strongly regular graphs are obviously metric. The line graph of thePetcr.un gr,tph and Uk lluffman-Singlet~xt graph an easy ez:unples of dixtance-regular graphc that are notstrongly regular.

Any k-regular graph of diameter dhas at most

-9-

ifktk(k-l)t . ~ ~ tk(k-1)d-~

veRices, as is easily seen. Graphs for whicó equality holds ate called Moore graphs. Moore graphs aredistance-regular, and those of diameter 2 were dealt with in Ttieorem 2.3. Using the rationality condiàons~AMFRIll. ~ lÓ~ and BANNN tiC. l7'O [2] tihOWed:

4.1. Theorem. A Muure graph with diameter d? 3 is u(2d t l)-gon.

A strong non-existence result of the same nature is the theorem of Fsrr 8c G. fitoAtAN [22] about finite

generalized polygons. A generalired m-gon is a point-line geometry such that the incidence graph is a con-nected, bipartite graph of diameter m and girth 2m. It is called regular of otder (s,t) for certain (finite orinfinite) cardinal numbers s, t if each line is incident with stl points and each point is incident with ttl

lines. (It is not difficult to prove that if each point is on at least three lines, and each line has at least tlueepoints (and m ~~), then the geometry is necessarily regular, and in fact s - t in case m is odd.) From sucha regulaz generalized m-gon of order (s, t), where s and t are finite and m 2 3, we can construct a distance-regular graph with valency s(ttl) and diameter d-[2m] by taking the collinearity graph on the points.

4.2. Theorem. A ftnite generalized m-gon of order (s,t) with s ~ 1 and t~ 1 satisfies m e(2,3,4,6,8 }.

Proofs of this theorem can be found in FFrr 8r HtcMArr [22], Kn.MO~x ~ So[.or.tox [36] and Roos [47];again the raàonality condiàons do the job. The Krein conditions yield some additional infotmation:

4.3. Theorem. A finite regulargeneralized m-gon wirh s ~ l ond t ~( satisfies s 5 tZ und t 5 sZ if m- 4

or 8; it satisfies s 5 t' and t 5 s 3 ifm- 6.

This result is due to Htct.ux [28, 29] and HwF~u.tts 8c Roos [26]. For each m e{ 2,3,4,6,8 } infinitely manygeneralized m-gons exist. (For m- 2 we have trivial structures - the incidence graph is complete bipartite;

for m- 3 we have (generalized) projecàve planes; an example of a generalized 4-gon of order (2,2) withcollinearity graph T(6) can be described as follows: the points are the pairs from a 6-set, and the lines arethe partiàons of the 6-set into three paits, with obvious incidence.)

Many associaàon schemes have the important property that the eigenvalues P;i can be expressed in

terms of orthogonal polynomials. An association scheme is called P-polynomial if there exist polynomialsfk of degree k with real ccefficients, and real numbets z; such that P;k - fk(Z; ). Cleazly we may always take

By the orthogonality relaàon ( 3.2.iii) we have

~k~fi (zr)Ít(zr) - ~ 4trP~iP;k - nnisik.

which shows that the fk are orthogonal polynomials.4.4. Theorem. An association scheme is metric if and only if it is P-potynomial.

ProoE Let the scheme be metric. Theorem 1.1 gives

A iAr -Pi; ~A~-i }PirA; }P~ii ~Arti.

Since p'~t~ x 0, A;t~ can be expressed in temts of A~, A;-t ard A;. Hence for each j there exists a polyno-mial fi of degree j such that

Ai -Ii(Ai).

Using this we have P;~E; - A~6; - fj(A ~)E; - fi(A i E;)B; - fi(P; ~)E;. hence P;i - fi(P; i).

Now suppose that the scheme is P-polynomial. "Itten the fi are orthogonal polynomials, and therefore theysaàsfy a 3-term recurrence relaàon (see Szr~ [52] p. 42)

ai,iÍi,i (z ) - (Q, -- )fi(- ) }Yi-ifi-i (z).

Herx~c

P;~Pr, - -a~tiP;i.i taiP~itYi-~P~i-~ fori - U....,d.

Since P; iP;i -~p~iP;; and P is nonsingulaz, it follows that p~i - U for ~l -j~ ~ 1. Now the full metric pro-~

perty easily follows by induction. ~

- lo-

Ttus result ís due to DlzsxxTE [l7] (Tbeotem 5.6, p. 61). There is also a restilt dual to this theorem,

involving so-called Q-polynomial and cometric schemes. However, just as the intersecáon ntunbets pj

have a combinatorial intetpretaáon while the Krein parameters q~ do not, the metric schemes have the

combinatorial descripáon of distance-tegular graphs. while there is no combinatorial interpretaáon for the

cometric property. For more information on P- and Q-polynomial associaáon schemes, see DII.swnTS [17]

and Bnn~~u 8c ITO [3]; for distance-tegular graphs, see the forthcoming book by Brouwer, Cohen 8c Neu-

maier.

The Hamming scheme and error-correcting codes.

Let X- Q~, the set of al vectors of Icngth d with entries in Q, where Q is some set of si-r.e q. Defirte

R; c X'- by (.r,y) E R; if the Hamming distance between x and y (i.e., the number of coordinates in which x

and y differ) equals i. This defines an associaáon scheme, the Hamming scheme H(d,q). The Hamming

scheme is easily seen to be metric, and hence by (4.4) P-polynomial. The orthogooal polynomiaLs involved

are the Krav~uk polynomials K~(x).

4.5. Theorem. For the Hamming scheme H(d,q) we have

P~i - Q;i - Ki(i ) - ~ (-1)k(q - IY ~(k)(i ~).k ~1

See DEts.4ttTS [17] (p. 38) or MncWu-t tavts 8t SLOnt.rE [43] for proofs. From (4.5) we see that P- Q, so the

Harttming scheme is self-dual.

A subset Y c X of H(d,q), such that YZ n R; - 0 for i- 1, -.., S-l and YZ n Ra ~ 0(i.e., a subset

Y such that the minimum Harrtming distance between two vectors of Y equals S), is nothing but an etror-

correcting code with parameters (d, ~Y~,S) over the alphabet Q(cf. the chapter of coding theory). Let

a-(a~ ~- u~) be the inner distribution of Y. Thcn ~a; - ~Y~, ao - l. a~ -''' -a~i -U. a 20 (bY

dclinition), :uxl uQ ? 0 by (3.7). Cunsidcr a~. ''. u~ as variablcs :uxt detinc,~ ,~

a' - l t max ~ a; subject to Ki( l) t ~ a;Ki(i ) 2 0. j- 0, .~., d, and a; 2 0, i- S. ~'~. d.

,-S ,35

Then clearly ~ Y ~ S a~. So a" is an upper bound for the number of codewords with a given length and

minimum distance. This bound, due to Dtl.snxTE [l7], is called the linear programming bound, since the

value of a' can be computed by linear programming. Of course, the above-menáoned inequalities are not

the only ones saásfied by the a;, and by adding extra inequaliáes to the system, one may obtain sharper

bounds on ~ Y~. For details and more applicatioos to coding theory, see DQ.sutTS [ l7], MnCWn r rwMC ~

St.ou.~ [43]. BFST, BROIIWFR, MACWII1IAMS, ODLYLRO áC SLOANE [6], BEST óC BROIJWER [S].

7'he Johnson scheme and t-designs.Let the set X consist of all subsets of size d of a set M, where ~M ~- m ? 2d. Define relation R; e XZ

by (x,y ) E R; if the Johnson distance between .r and y(i.e., the cardinality ofx`y) equals i. 'Iitis defines an

association scheme, the Johnson scheme J(d,m). Since the Johason distance between x and y equals twice

the H:tmming distance between ( the charactetistic vectors of) .r and y, it follows that also the Johnson dis-

tance satisfies the [ri:tngle ineyuality, so that the Johnson scheme is metric. Note that the graph (X,R t) is

complcte for d- 1, and is the tri:utgular graph T (rn) for d- 2. 'ilte following result (due to Oo~swwnuw

~46j and YnM,~MnTO et a. [54~) givcs some par:uneters:

4.6. Theorem. Fur the Juhnsun scherrre J(d,m ) the frdfuwing huld:

n. i d-k d-i m-Ji,t-i c~ i J-i in~-i

Pri - ~ Q~; - Ei(i) - F,(-lY-k~l-k)( k )( k ) - `(-1)k(k)(~-kX j-k )~il; k~ k~0

iti-(~)-(imt). nJ-(~)(m~).

Here E~(x) is a so-called Eberlein polynomial. It has degree 2j in the indetemtinate .r, and degree j in the

iodeterminate z(mtl-x). Since Pi1 -d(m-t)-i(mtl-i), E~(i) indeed has degree j in P;t as required by

the definiáon of P-polynomial scheme.

-ll-

The graph ( X,Rd) of a Johnson scheme is called a Kneser graph. A subset Y of X such that any twoelements of Y have non-empty intersection is a coclique ( independere seq of the Kneser graph. By use ofTheorem 3.7 and the above formulas, it can be deduced that

m-1~Y~ S~d-t)~

the famous result of Etu~s, Ko 8t Rwno [21 J.

A t-(m,d,7~) design is a subset Y c X of the Johnson scheme J(d,m) such that eacó t-element subset

of M is contained in precisely l elements of Y. Da.swalE [ 17J (Theorem 4.7, p. S 1) proved the following:

4.7. Theorem. A nnn-empty subset Y of the Johnson scheme J(d,m) with inner distributiona-(aa ~-, ad) is a t-(m,d, ~,) design if and only if

e~a;Q;i-O forj-l,~~.,t.~~

Just as we did in case of the Hamming scheme, we can define

(;)

d

a. - I tmin~a; subject to,-I d d

a; ? O for i- 1, ~., d, and ~a;Q;i - 0 for j- 1, -.~, t, and ~a;Q;i ? O forj- t tl, .~~, d.;~ r~

where the Q;~ are found in (4.6). Now we have the linear programming (lower) bound for the number of

blocks in a rdesign: ~Y~ ? a.. Using the simplex algorithm, HwFa,ff.tu ~ WEUC [27] showed that this ine-

quality implies the noo-existence of the designs with parameters 4(17, 8.5), 4(23, l 1,1t.), ~, - 6,12, 4-

(24,12,15 ), 6-( l9, 9, ~.), ~. S l0. 6-(20,10, ~,), ~. - 7, l4. In certain other cases, such as 5-(19, 9, 7), KSw~x

[34] ruled out the existence of t-designs by showing that no soluàon of the above system of inequalities

corresponds to an actual design.

[f we replace the restricàons (aQ); - 0 by restrictions a; - 0( l 5 i ~ 2S) in this system of inequali-

ties, and maximize ~,a;, we obtain upper bounds for the caníinality of codes with minimum distance S andconstant weight d.

We might also require both a; - 0 and (aQ)~ - 0 for suitable i, j, and obtain results for t-designs with

restricted block intersecàons, such as quasisymmetric block designs. By this method it follows for instance

that a quasisymmetric 2-(29,7,12) design with block intersecàons L and 3 cannot exist (Hwen~xs [24]). For

details and more results we iefer to DQ.swx~ [17], MwcWu.LtwMS ~ SLOwxE [43], Brsr, BROUwex,

MACWII,LL4MS, ODLYLRO áL SLOANE [GJ, CM~ff.RON BC VAN LIN'C [1SJ, Cw~~Qawrrtc [l3]. We also point out

that several n:sults of ~3 in Godsil's chapter can also be obtained by use of the framework of Johason

schemes.

Dil.swxr6 [ 17J generalized the noàon of t-designs to subsets of arbitrary association schemes saàsfy-

ing ('). (Equivalently, a t-design is a subset Y of X such that its characterisàc vector x saàsfies E~x - 0 for

j- 1, ~~ , r.) He shows that a t-design in the Hamming scheme is what is known as an orthogonal array of

strength t(see the chapter on Block Designs). 'Ittus, we also have a litxar programming bound for orthogo-

nal arrays. More generally, one may give an interpretaàon of the classical concept of t-design in terms of

ranked posets in the obvious way, and then prove for eacó of the eight known infinite families of P- and

Q-polynomial associaàon schemes that a subset is a classical t-design if and only if it is a Delsarte f(t)-

design (where usually f(t) - t), see DE.xs.vtTE [19] and SrwNrorr [S1].

Imprimitive schemes.

In krtion Z we have seen a c:umpletcly diffemnt rclation Ixtwecn design.i arxt asuxiation sctxnks.

l.et us givc orx: morc example. L.et N bc tlx; incidencc matrix of a squ:ue 2-clcsign. Then, defining Ao - l,

A,-INr ~J . A,-I~~Jand A3 -J-I-AI~ -A,. we obtainl a 3-class association scheme. It is imprimitive, that is, the union of

some of the R; form a non-trivial equivalence relation (here RZ is an equivalence relation). Another

-12-

imprimiáve association scheme we óave seen is the lice graph of the Petersen graph (thete having mazirttaldistance is an equivalence relation).

Given an imprirniáve associaáon scheme one may produce new association schemes; on the onehand, there is a natural way to give the set of equivalence classes the swcture of an association scheme(the `quotient scheme'), and on tbe other hand, eacó equivalence class together with the restricáons of thenriginal relations becomes an association scheme (a 'subscheme' of the original scheme).

The group case.

We very brietly discuss some mlations between associaáon schemes and fmite perntutaáon groups.

Let G be a permutation group acting on a set X. Then G has a natural acáoo on X2, the otbits ofwhich are called orbitals. Suppose G acts generously transitive on X, that is, for any x, y e X there e~tistsan element of G interchanging x and y. Then the orbitals form an associaáon scheme.

(Without any requirements on G, the orbitals fonn a coherent configuraáon (see ~ 3). The coherentconfiguraáon is homogeneous if G is áansitive. We get an association scheme in the sense of Delsartewhen the permutaáon character is muláplicity free.)

For any x e X, the number of orbitals equals the number of otóits on X of G, (the subgroup of G ofpennutaáons fixing x). This number is called the rank of G. Thus, the number of classes in the associaáonscheme is one less than the rank of G. We can also transfer other pertnutaáon group theoreác terminologyand results to the tóeory of associaáon schemes. For instance, the Bose-Mesner algebra is in the groupcase known as the centralizer algebra, and all standard results on this centralizer algebra (cf., e.g.,W~.nrrar [53]) have their direct analogue for the Bose-Mescer algebra.

The Hamming and Johnson schemes are derived from generously transitive permutation representa-tions as discussed above; for instance, the Johnson scheme is derived from the representaáon of the sym-metric group Sym (m) on the d-element subsets of an m-set.

If a metric associaáon scheme belongs to the group case, then the corresponding distance-regulargraph is called distance-transitive. In other words, a graph is distance-transiáve when its group of automor-phisms is transitive on pairs of vertices with a given distance. Distance-transitive suongly regulaz graphs

are known as rank 3 graphs. A rank 3 permutaáon group is generously transiáve if and only if it has evenorder, consequendy every rank 3 pennutaáon group of even order provides a strongly regular graph. Allsuch strongly regular graphs have recendy been classified, see Kwrrroe 8c L~st.r~ [35], L~sECx [38,40],and L~aECx 8c Swia. [39].

References

1. Aschbacher, M., The non-ezistence of rank three permutation groups of degree 3250 and subdegree57, J. Algebra 19 (1971) 538-540.

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- t3-

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i

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203 Antoon van den 61zen and Dolf TalmanA new strategy-adjustment process for computing a Nash equilibrium ins noncooperative more-person game

204 Jan VingerhoetsFabrication of copper and copper semis in developing countries. Areview of evidence and opportunities

205 R. Heuts, J. van Lieshout, K. BakenAn inventory model: what is the influence of the shape of the leadtime demand distribution?

206 A, van Soest, P. KooremanA Microeconometric Analysis of Vacation Behavior

20~ F. Boekema, A. NagelkerkeLabour Relations, Networks, Job-creation and Regional Development. Aview to the consequences of technological change

208 R. Alessie, A. KapteynHabit Formation and Interdependent Preferences in the Almost IdealDemand System

209 T. Wansbeek, A. KapteynEstimation of the error components model with incomplete panels

210 A.L. HempeniusThe relation between dividends and profits

211 J. Kriens, J.Th. van LieshoutA generalisation and some properties of Markowitz' portfolio selecti-on method

212 Jack P.C. Kleijnen and Charles R. StandridgeExperimental design and regression analysis in simulation: an FMScase study

213 T.M. Doup, A.H. van den Elzen and A.J.J. TalmanSimpliciel algorithms for solving the non-linear complementarityproblem on the simplotope

214 A.J.W. van de GevelThe theory of wage differentials: a correction

215 J.P.C. Kleijnen, W. van GroenendaalRegression analysis of factorial designs with sequential replication

216 T.E. Nijman and F.C. PalmConsistent estimation of rational expectations models

ii

21~ P.M. KortThe firm's investment policy under a concave adjustment cost function

218 J.P.C. KleijnenDecision Support Systems ( DSS), en de kleren van de keizer

219 T.M. Doup and A.J.J. TalmanA continuous deformation algorithm on the product space of unitsimplices

220 T.M. Doup and A.J.J. TalmanThe 2-ray algorithm for solvíng equilibrium problems on the unitsimplex

221 Th. van de Klundert, P. PetersPrice Inertia in a Macroeconomic Model of Monopolistic Competition

222 Christian MulderTesting Korteweg's rational expectations model for a small openeconomy

223 A.C. Meíjdam, J.E.J. PlasmansMaximum Likelihood Estimation of Econometric Models with RationalExpectations of Current Endogenous Variables

224 Arie Kapteyn, Peter Kooreman, Arthur van SoestNon-convex budget sets, institutional constraints and imposition ofconcavity in a flexible household labor supply model

225 R.J. de GroofInternationale cotirdinatie van economische politiek in een twee-regio-twee-sectoren model

226 Arthur van Soest, Peter KooremanComment on 'Microeconometric Demand Systems with Binding Non-Ne-gativity Constraints: The Dual Approach'

22~ A.J.J. Talman and Y. YamamotoA globally convergent simplicial algorithm for stationary pointproblems on polytopes

228 Jack P.C. Kleijnen, Peter C.A. Karremans, Wim K. Oortwijn, WillemJ.H. van GroenendaalJackknifing estimated weighted least squares

229 A.H. van den Elzen and G. van der LsanA price adjustment for an economy with a block-diagonal pattern

230 M.H.C. PaardekooperJacobi-type algorithms for eigenvalues on vector- and parallel compu-ter

231 J.P.C. KleijnenAnalyzing simulation experiments with common random numbers

iii

232 A.B.T.M, van Schaik, R.J. MulderOn Superimposed Recurrent Cycles

233 M.H.C. PaardekooperSameh's parallel eigenvalue algorithm revisited

234 Pieter H.M. Ruys and Ton J.A. StorckenPreferences revealed by the choice of friends

235 C.J.J. Huys en E.N. KertzmanEffectieve belastingtarieven en kapitaalkosten

236 A.M.H. GerardsAn extension of KtSni.g's theorem to graphs with no odd-K4

23~ A.M.tL Gerards and A. SchrijverSigned Graphs - Regular Matroids - Grafts

238 Rob J.M. Alessie and Arie KapteynConsumption, 5avings and Demography

239 A.J. van ReekenBegrippen rondom "kwaliteit"

240 Th.E. Níjman and F.C. PalmerEfficiency gains due to using missing data. Procedures in regressionmodels

241 Dr. S.C.W. EijffingerThe determinants of the currencies within the European MonetarySystem

1V

IN 198~ REEDS VERSCHENEN

242 Gerard van den BergNonstationarity in job search theory

243 Annie Cuyt, Brigitte VerdonkBlock-tridiagonal linear systems and branched continued fractions

244 J.C. de Vos, W. VervaatLocal Times of Bernoulli Walk

245 Arie Kapteyn, Peter Kooreman, Rob WillemseSome methodological issues in the implementationof subjective poverty definitions

246 J.P.C. Kleijnen, J. Kriens, M.C.H.M. Lafleur, J.H.F. PardoelSampling for Quality Inspection and Correction: AOQL PerformanceCriteria

24~ D.B.J. SchoutenAlgemene theorie van de internationale conjuncturele en struktureleafhankelijkheden

248 F.C. Bussemaker, W.H. Haemers, J.J. Seidel, E. SpenceOn (v,k,a) graphs and designs with trivial automorphism group

249 Peter M. KortThe Influence of a Stochastic Environment on the Firm's Optimal Dyna-mic Investment Policy

250 R.H.J.M. GradusPreliminary versionThe reaction of the firm on governmental policy: a game-theoreticalapproach

251 J.G. de Gooijer, R.M.J. HeutsHigher order moments of bilinear time series processes with symmetri-cally distributed errors

252 P.H. Stevers, P.A.M. VersteijneEvaluatie van marketing-activiteiten

253 H.P.A. Mulders, A.J. van ReekenDATAAL - een hulpmiddel voor onderhoud van gegevensverzamelingen

254 P. Kooreman, A. KapteynOn the identifiability of household production functions with jointproducts: A comment

255 B. van RielWas er een profit-squeeze in de Nederlandse industrie?

256 R.P. GillesEconomies with coalitional structures and core-like equilibrium con-cepts

V

257 P.H.M. Ruys, G. van der LaanComputation of an industrial equilibrium

Bibiiotheek K. U. Brabant

Y II~ I III I II i i i i~ ~i i1 7 000 O ~ 059992 7