Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.

In Chapter 7 we will look at the standard representations of distance-regular graphs. In the first section we will develop some theory using ideas of Terwilliger and Godsil. In the second section we will apply this theory to distance-regular graphs with ad = 0. This section is based on [J]. In the last section we will give the ciassification of distance-regular graphs with an eigenvalue of multiplicity 8. This section is joint work with W.J. Martin.
At the first place 1 would like to thank my supervisor Andries Brouwer for his continuous support and patience during the period of my research. Furthermore 1 would like to thank the Discrete Mathematics group for the stimulating working environment, especially I am grateful to Aart Blokhuis, Hans Cuypers and Henny Wilbrink. Also my thanks goes to prof.dr. J.J. Seidel for numerous discussions and for the careful reading of (parts of) a first draft of my thesis.

Introduction
In this chapter we will introduce distance-regular graphs. Also we will expla.in the subjects of this thesis. There is only one exception, the metric theory of graphs will be introduced in Chapter 6. Only Section 3 is new in this chapter. For the notations we follow [19].
for i, j, t = 0, 1, ... , d. In particular the pfj are non-negative integers. Now we look at the eigenvalues of r. The adjacency matrix A has at least d + 1 different eigenvalues, since the matrices A; = v;(A),(i = 0,1, ... ,d) are linearly independent.
But also AAd = bd-1Ad-1. + adAd, so there is a polynomial w of degree d + 1 such that w(A) O; therefore A has exactly d + 1 distinct eigenvalues.
Define the tridiagonal ( d + 1} x ( d + 1) matrix The algebra.ic multiplicity is equal to the numerical multiplicity for all the eigenvalues of L 1 , because tridiagonal ma.trices are similar to tridiagonal symmetrie matrices.
For all numbers r E R the matrix L 1 -r I has rank at least d, because c; :f. 0, Let x E vr. Define the vector w by Wy u; ( 8) if d( x' y) = i. It is easy to see that the vector w is an eigenvector of A corresponding to fJ. The conclusion is tlmt the eigenvalues of r can be calculated from the intersection arra.y only. Also the multiplicities of these eigenvalues can be calculated from the intersection array only, cf. [19,Theorem 4.1.4]. The array tt; ( 8) is called t11e stamlmYl se<1uence corresponding to 8. Now we give some easy restrictions on the intersection arrays for distance-regular graphs. (iii) IJ i + j $ d, then c;:::; bi.
( v) All the multiplicities of the eigen11alues are integml. D lndeed, Pl; and multiplicities can be calculated from the intersection array. The significa.nee of this proposition is that it is a sieve for potentiaJ intersection arrays. An intersection array is sometimes called 'feasible' if it passes these five tests. Thus, the intersection array of a distance-regular graph is feasible, hut a feasible array need not correspond toa distance·regular graph. If there are induced quadrangles in the distance-regular graph, theu T('rwilliger had shown the following theorem. (A stmngly regufor grnph is a distance-regula.r graph of diameter 2. In the next section we sec some properties of strongly f('gular graphs.) 3 Theorem 1.2 (TERWILLIGER [106,104], cf. ( (i) The complement of the triangular graph T(7) = J (7,2). It has 21 vertices and intersection 01-ray {10,6; 1,6}.

Strongly regular graphs and related graphs
A gra.ph of dia.rnet<'r 1 is a clique and hence distance-regula.r with intersection array { k; 1 }. A counected strongly regular gra.ph is a distance-regular graph of diameter 2.
If it has intersection arra.y { k, b 1 ; 1, c2} then we say that it is a strongly regula.r graph with para.meters ( v, k, >.., JL ), where v is the number of vertices, À = ai and µ = C2. The following t.heorem gives some properties of strongly regular gra.pbs. (iii) The eigcrwalue.• r mul s are integers, unless r and s hat1e the same multiplicity.
Theu r, s = (-1 ± ,/V)/2 and r is called a conference gra.ph. Conference graphs haue pammetCl"s ( v, 1.:, >.., 11) = (4t + 1, 2t, t -1, t) /or some t ?. 1. IJ Jt < k tlien the complement of r is again a connected strongly regular graph with parameters ( v, k, X, µ;), mhere k = v -k -1, ïi= v 2k->.., and has eigent)(llttes k, -.~ -1, -r -1. D Exa.mples of conference graphs are the Paley graphs QR(q), defined for prime powers q = l(mod 4), which have Fq as vertex set and where two vertices a.re adjacent if and only if their difference is a. nonzero square, cf. PALEY (87]. (QR is mnemonic for 'quadratic residue'.) A generalisa.tion of strongly regular graphs are amply regular graphs. A connected gra.ph is a.mply regular with parameters ( v, k, >..,µ) if it is regular with va.lency k, ea.ch edge lies in >.. triangles and each pa.ir of vertices at distance 2 have µ common neighbours. We will see that if µ ;:: 2 then the regula.rity condition is superfluous. A generalisation of the a.mply regula.r graphs are the ( s, c, a, k )-grap hs. For any integers s, c, a, k with a + 2, s, c, k ;:: 2, a graph r is called an ( s, c, a, k )-graph if it satisfies the following properties: (i) the girth of r is 2s jf a = 0, and 2s 1 otherwise, (ii) for two vertices at distance s -1 there are a paths of length s connecting them, (iii) for two vertices at distance s, there are c pa.tbs of length s connecting them, (iv) the maxima.l valency in ris k.
They were introduced by TERWILLIGER [103]. He showed the following proposition.

U niformly geodetic graphs
A uniformly geodetic graph is a connected graph sneb tha.t the number of geodesics (i.e. shortest pa.tbs) between a.ny two vertices x a.nd y only depends on the dista.nce d(x,y).
These graphs were introduced by CooK AND PRYCE [32). They are a.lso called Fgeodetic graphs, see CECCHERINJ AND SAPPA [27] and SCAPELLATO [92], where F(j) is the number of geodesics bet ween two vertices at distance j.
For x, y vertices at distance i in a graph r we define c;(x, y) = f;_ 1 (x) n f(y), a;(x, y) = f;(x) n f(y), b;(x, y) = f;+i (x) n f(y). We say that the number c; (resp. a;, b;) exists if c;(x,y) (resp. a;(x,y), b;(x,y)) does not depend on the vertices x,y.
The foJlowing lemma gives a trivia.l conscquence of this dcfinition. Lemma 1.6 A connected graph is tmiformly geodetic if and only i/ the nttmbers c; exist. 0 Thus uniformly geodetic gra.phs genera.lise distance-regular graphs and trees. An easy property of uniformly geodetic graphs is the following. Lemma 1.7 Fora tmiformly geodetic gmph wilh numbers c;, i 1, 2, ... ,d we have 0 In the next subsection we will characterise bi1>a.rtite uniformly geodetic graphs and give some examples of them.
(ii) IJ q > 1, then ris a Gq(n,t,t+ 1). o This theorem generalises a result of RAY-CHAUDHURI AND SPRAGUE [88]. Now  The partit.ion into singletons is a.Jways regular; the pa.rtition {Vf} is regula.r predse]y when r is regular. For any group of automorphisms of r, the pa.rtition of Yf into Gorbits is regular. We already have seen that the partition { { x}, r 1 ( x ), r 2( x ), ..• , r d( x)} of the vertex set of a distance-regular graph r, where x is a vertex of r, is regular.
Regular partitions give information about the spectrum: the eigenvalues of the matrix (eAB)A,BeP are eigenvalues of r (with at least the sa.me multiplicity). Much more information can be found in GoDSIL & McKAY [50] (who, following SCHWENK [96], call such partitions equitable).
The distribution diagram of r with respect to a regular partition P consists of a number of balloons bA, one for each element A E 'P, and a number of lines LAB(= LBA) joining two balloons bA and bB. one for each pair {A, B} for whkh EAB f:. 0 (then also EBA f:. 0).
The lines LAA are usually not drawn. This diagram is provided with numbers as follows: in the balloon bA we write IAI, and at the A-end of LAB we write EAB· The number EAA is just written to bA; som~times, when EAA = 0, we write '-'.
Examples. The structure of the dodecahedron around a face is shown by 2 2 We obtain amore complica.tecl picture by looking at the structure around the two vertices x, y of an edge in the Petersen graph 1 x: 1. y: As suggestecl by the two examples, we want to determine P by giving a few of its elements. And indeed this ca.n be clone in a canocical way: Proposition 1. 16 Let S be an arbitrary partition of the vertex set Vf of a gmph r. Then there is a tmiqtte coarsest partition P of vr finer than S, that is reguLar in the above sense. D Thus, we have a.ssocia.ted a distribution dia.gram with a.n arbitra.ry pa.rtition of Vf; the 'distribution diagram around C' for Ca. subset ofVf is that associated with {C, Vf/C}.

Codes and completely regular codes
For us a. code in a graph r is just a. non-empty subset of the vertex set of r. Let C be a code in agraph r. Fora vertex x, the distance d(x,C) is defined by d(x, C) = min{d(x,c) 1 c E C}. Define C; = {x 1 d(x,C) = i}. We denote the cardinality of C; by K;. The covering radius, p, of a code is the maxima! m such that Cm f:. 0.
A code in a connected graph is called completeLy regttlar if the partition 8 is regular. We denote by "ti (resp. a;,/3;) the numher ec,,ci-l (resp. ec;,C;,ec;,C;+i). A completely regular code has intersection diagram: ap A completely regular code in a regular graph has intersection array Remark that then a; = k -{3; -7;, if kis the valency of r. This definition of completely regular codes is due to Neuma.ier and he has shown that this definition is equivalent with the original definition of completely regular codes in distance-regular gra.phs by Delsarte.
Note that for a distance-regular graph and a distance-hiregular graph each vertex is a completely regular code. It was shown by GoosIL AND SHAWE-TAYLOR [51) that all connected graphs such that every vertex is a completely regular code are distance-regula.r or distance-biregular. Remark that completely regula.r codes come in pairs. A code C is completely regular if and only if C P is completely regula.r. It is not true tha.t for a completely regular code in a dista.nce-regular gra.ph, the array 7 1 , 1 2 ••• 7 P is an increasing array. We saw a.lrea.dy that the penta.gon in the dodeca.hedron is a.n exa.mple for this. In Chapter 3 we will show that this array is an increa.sing array for most classical gra.phs. Also in tha.t cha.pter we will show that the array Ko, K 1 , ••• , Kp is not always unimodular for a. completely regula.r code in a. dista:nce-regula.r gra.ph.
A code C is a perfect (e-error-correcting) code if for each x E Vr there is a unique c E C with d(x,c) ~ e. Note that a perfect e-error-correcting code is completely regular in a. distance-regular graph and has intersection array { k, bi, . .. , be-l; ei, c 2 , ••• , c.}. A partition P is called completely regular if it is regular, and all C E P are completely regular with the same intersection diagram. Note that a completely regula.r partition is uniformly regula.r. Let P be a. partition of the vertex set of r. The qttotient graph r /P is the graph witl1 vertices the classes of P and two classes C, D are adjacent if dr(C, D) = 1. The following theorem shows tha.t if you have a completely regula.r partition Pin a distance-regular graph, then the quotient graph is a.lso dista.nce-regular. Theorem 1.17 ([19), Theorem 11.1.6.) Let P be a uniformly regular partition of a distance-regular gmph r. Then r /P is distance-regttlar i/ and only ij P is completely Let Che a. linea.r code over Fq (i.e. a vector spa.ce over F 9 ). We consider Cas a subset of the vertex set of the Ha.mming gra.ph H ( n, q ). If C is completely regular in H ( n, q ), then also all its translates C + c, c E F~ are completely regular with the sa.me pa.ra.meters as C. Hence the quotient graph H(n,q)/P is distance-regula.r, where Pis the pa.rtition of F~ in to the cosets of C. In this case the quotient gra.ph is ca.lled the coset graph belonging to C.
Example. The binary Golay code is perfect a.nd linear, hence t11e coset graph r belonging to this code is distance-regula.r and has intersection array {23,22,21;1,2,3}. 9 On subgraphs in distance-regular graphs

Pappus subgraphs
In this section we give a sufficient condition to asure tha.t there Pa.ppus subgraphs in a dista.nce-regnlar graph. As an applica.tion of this we are able to rule an array as the intersection array of a distance-regula.r graph.
By c3 = 2 we have <l(zi, z4) = 4 (the distance is 2 or 4, hut not 2, since in tha.t case there would be more tban two shortest pa.tbs joining z 4 and y 4 which are at dista.nce 3 from each other). By looking at the geodesics between y 1 and Y6, there is a vertex Ut such that z 1 "' Ut "" z 6 and a. vertex u 2 such that z 2 "' u 2 "" z 4 • In the same way, we have a vertex u 3 with z 3 "' u 3 "" z 5 • All these u; are different. By looking at the pair ( zi, z4) one can see tha.t there is a vertex Vt with u1 "' v1 "' u2. Simila.rly, there is a. vertex V2 with ttt "' V2 "' 1'3.

.2.1 Introduction
In this section we stttdy distance-regular subgraphs of distance-regular graphs. In the second subsection we give some sufficient conditlons to assure that the graph induced by the geodesics between two vertices is distance-regular. TERWILLIGER [103] has given the diameter bound d :;5 (sl)(k-1) + 1 for distanceregula.r graphs with girth 2s and valency k ;:::: 3. In the third subsection we show that the only distance-regular graphs with even girth which reach, this bottnd are the hypercubes and the doubled Odd graphs (Theorem 2.7) and give a somewhat improved diameter bound for bipa.rtite distance-regular graphs.
In the fottrth subsection we study distance-regular subgraphs in a hypercube. In this subsection the subgraphs, are not necessarily subgraphs. WEICHSEL [109] has studied them and conjectured that the only distance-regular subgraphs of a hypercube are the even polygons, the hypercubes and the doubled Odd gra.phs and proved this in the case of girth 4. We show that if the girth is 6, then it must be a doubled Odd graph (Theorem 2.14). If the girth is eqttal to 8 then the valency is at most 12 (Theorem 2.17).

Substructures
I,et f be a. gra.ph. For two vertices x, y of r, put C(x, y)  Let ), hut these two sets have the sa.me ca.rdinality and thus they are equal.

Remark 1. For c;
i, MULDER [80,81] has shown the previo11s theorem without assumption (ii). More examples are given below. Proposition 2.5 If r is the collinearity graph of a near polygon then ( ii) holds.

13
Proof. By the previous theorem and Proposition 3, for any pair of vertices.x,y at distance 2i + 1 the subgraph induced by C(x,y) is a bipartite distance-regular graph with k = 3, c; = 1, Ci+I = .. . C2i = 2 and C2i+1 = 3. By DAMERELL [34] and also by BANNAI & ITo [11] there are no Moore graphs with diameter at least 3 and valency at least 3.
Proof. We calculate the number of vertices z on level j with d( z, y) = ij. This number equals the right side of (2.1 ). Suppose tha.t precisely ei -s neighbours of y on level i -1 (considered as (i -1)-sets) contains the vertex z ( considered as a j-set ). Then 0 :5 s :5 e a.nd we find that the number of vertices z is at most the left hand side of (2.1). D The proof of WEJCHSEL [109], Theorem 5 shows Lemma 2.11 Let r be a distance-reg11lar subgraph of a cube with valency k and girth 2t ;::: 6.
(i) IJ v is a vertex of r on level r and of weight r, then 2cr-l +er S 2r -1.
Lemma 2.12 Let r be ei distance-regular subgraph of a cube with valency k and girth 2t.
Proof There are no bipartite distance-regular graphs with the parameters· of cases ( ii), ( iii) and (iv).
If k = 3, then we have the following possibilities.

Completely regular codes
In the first three sections of this chapter we study a conjecture of MARTIN [74) on the parameters of completely regular codes in distance-regular graphs. In Section 1 we give some simple properties of completely regular codes. In Section 2 we show that this conjecture is not true in general, hut for most classical graphs it is true. In Section 3 we show that there is a counterexa.mple in the Biggs-Smith graph for a weakened version of Martin's conjecture. Furthermore in this section we classify the completely regular codes in the Biggs-Smith graph.
In Section 4 we study perfect codes in the Hamming scheme with two protective radii. We especially are interested in the subclass of bipartite perfect codes. This subclass is also a class of completely regular codes in the Hamming scheme. The ma.in goal of this section is to give a non-existence result on this subdass of perfect codes. The first three sections a.re based on the preprint of the author "On a conjecture of Martin on the parameters of completely codes and the dassification of the completely regular codes in the Biggs-Smith grapb", (68]. The fourth section is based on the paper with J .M. van den Akker and R.J.M. Vaessens, "Perfect codes with distinct protective radii", [2]. First we need some definitions and notation. Let C be a subset of the vertex set of r.  [74], Theorem 2.4.6). Let r be a distance-regular antipodal cover of a distance-1Y?gular· gmph A and let 11' : r ..... A be the projection. Then /or each completely regular code C in A the code D = ir-1 (C) is a completely regular code in f with the same intersection array as C. D This proposition has a partial converse. Theorem 3.8 Let r be a distance-re911lar antipodal 2-cover of a distance-regular graph A. Let x' be the antipodal of x in r. Let C be a completely regular of r with covering mdius p. Then eithe1' .1: There is ani such that x 1 E C;. Now by Theorem 3.5 for all y E C we have y' '/. C. Let I ={il 3yec(y' E C;)}. Again by Theorem 3.5 we have U;E[C; = {y'ly E C}. Now since {y'ly E C} is a completely regular code with the same parameters as C, it fo\lows that its covering radius is p. Consequently C P = {y'IY E C}. D Remark 1. The above theorem is a generalisa.tion of the fact that in a distance-regular 2-cover fora perfect code C one has x E C if and only if x' E C, cf. [19], Rema.rk on the bottom of p. 439.

On a conjecture of Martin
In [74], Martin conjectured that "li :S "li+i and /J; !! /Ji+i for a completely regular code in a distance-regular graph. In this section we prove this conjecture for several families of distance-regular graphs and give a family of counterexamples in the doubled Odd and Odd graphs showing that it does not hold in general. Even a weakened version of this conjecture is not true. It is not true that the numbers K; of a completely regular code in a distance-regular graph forma unimodal array. We shall give an example in Section 3.3.1. A very simple counterexample to Martin 's conjecture is the following one. Take five vertices in the dodecahedron Li with induced subgraph a pentagon. This is a. completely regular code in Li with intersection diagram 2 2 and we see that Î3 < 7 2 , fJo < /31. Below we give an example where the array {"!;}; is not unimodaJ.
Lemma 3.9 Let r be a connected gmph. Let C and D be two completely regular codes in r with the same parameters. Lets:= max{i 1C;nDf.0} Then c;(C) :S Ci+.(C) and Then y E C;+•-1 by the triangle inequaJity. Hence c;+.(C) !! c;(D) and thus c; The proof for the second inequality is analogous. D The above example also shows that this lemma becomes fälse if we replace s by s -1 in the inequalities. Proof, By the fa.et that 4> is an automorphism, it follows that </>( C) is a completely regular code in f with the sa.me parameters as C. By the previous lemma we are done.

D
For a graph f, denote by Ï' the graph whose vertices a.re the symbols x+, x-, x a vertex off, and whose edges are the 2-sets {x",y"}, with x y or x"' y, and tT f:: r.
The graph Ï' is called the extended hipartite double of r. Proof. The fact that f is a extended bipa.rtite double of D. is equivalent with tha.t r is bi parti te and has a.n automorphism </>of order 2 such tiiat </>( x) "" x for all vertices x of r and f/</> ~Ll. Let C 1 he acompletely regularcodein r. Suppose that </>(C 1 ) C 1 • Then  Proof. Only symmetrie bilinear forms graphs with diameter a.t least 3 and characteristic odd, or with diameter at least 4 and characteristic even are not distance-regular, all the other graphs are distance-regular, cf. [19], p. 286 and the errata of that book. The gra.phs of families (xiv), (xviii), (xx) a.nd (xxi) are not flag-transitive. The other graphs are flag-transitive and by Lemma 3.11 for any graph not a member of family (xiv), (xviii), (xx) or (xxi), we only have to find an automorphism </>,not the identity, such that d( </>( x ), x) ~ 1 for all vertices x. Let r be a graph of one of the above families. If ris a Hamming graph then you may take for </> a translation over a one-weight vector. If r is a Johnson graph then </>is a transposition. If r is a Grassmann graph then </> is a transvection. If r is a symplectic dual polar graph then </>is a symplectic transvection. H ris an orthogonal dual polar graph and the characteristic is not two then </>is a reflection. Proof. Let C be a cmnpletely regular code in r. If r is a doubled Odd or Odd graph and suppose that thcre is a vertex i: (j; C. Then there is a transposition </> such that <f>(C) -:f G and by Lemma 3.10 wc are done.
If ris a doubled Grassma1m graph and suppose that there is a vertex x </: C. Then there is a transvection </>sneb that <f>(C) # C and a.gain by Lemma  Now the parameters follow easily. D For n ;::: 3 and i = 1 you get /32 > f31. The array K; is decreasing for C. It is an open question whether the array x:; of a completely regular code in a doubled Odd graph has to be unimodal. For all c E C we have N \ c E C and this implies that C is a completely regular code in 0"+1 with the same intersection array.
For n = 7 and i = 3 you get î't = 4, î'2 = 6, î'J = 5, 1' 4 = 7 and hence the array {î';}; is not unimodal. There is still no example known of a completely regular code in a doubled Grassmann graph with "'(; > "Yi+t. hut I conjecture that there must be such an example. to see that ' ljJ is an automorphism of r of order 8. The group K =< rr, ' ljJ > is isomorphic to 17: 8 (this is the semi-product of a C11 and a C 8 , using the notation of [30]). Now K bas two orbits on r a.nd one of them is X U Y. By the fact tha.t K is a maximal subgroup of PSL (2,17) it follows that Il is the stabiliser of X U Y and of Z1 U Z2 U-Z4 U Zs. It is obvious that ~, 4 is the only nou-identity element of < t/J > which preserves the partition fl = { X, Y, Z 1 , Z 2 , Z 4 , Zs}. Also obvious is the fact that < ef; 2 > is the maximal subgroup of< t/J > which preserves the partition fl 1 = {X, Y, Z 1 U Z 4 , Z 2 U Z 8 }. In the same way you see that only the ideutity of < t/; > stabilises Z 1 U Z 2 . This is also the only element of < 'ljJ > wllich stabilises Z1 U Z 8 • So we have shown the following proposition. ( iii) The stabiliser of the partition {X U Y, Z 1 U Z2, Z4 U Zs} is isomorphic to D34 and has six orbits on r.

Completely regular codes in the
( iv) The stabiliser of the partition { X U Y, Z 1 U Zs, Z4 U Z2} is isomorphic to D34 and has six orbits on r.

Classification of the completely regular codes in the Biggs-Smith graph
In this snbsection we dassify the completely regular codes in the Biggs-Smith graph. We say that a. completely regular C with covering radius p in a distance-regular graph À is trivial if IC! :S 1, ICPI :S 1 or all the vertices of À are in C. Let   The minimum distance of C is less then 6. Proof. Snppose that the minimum dista.nce is 7. By [74], Theorem 2.3.3 it follows that ICI > 2. Because of Pf. 1 = 1 we get ICI = 3. But it is easy to see that such a code is not completely regular. Suppose uow tha.t the minimum distance is 6. Then we have /30 = 3, /31 f:h = ")'3 = 2, /1 1'2 = 1 and <to = a1 a2 = 0. If a3 :/= 0 then a3 = 1 and Lf=o lii = 16Ko, hut 16 does not <livide 102. So a3 = 0. By calcula.ting the number of 9-cycles in through one vertex we get that this number is at most /Jo/11/32/33 < 24, because there are no 6-, 8-and 10-cycles in r. But the number of 9-cycles through one vertex in r equaJs 24. 0 Lemma 3.22 The minimum distance of C is not equal to 5.
Proof. Suppose that the minimum distance is 4. Then a 1 = a 0 = 0, {3 0 = 3, {3 1 = 2 = î'z and î't 1. In r every two vertices at distance 4 there is a path of length 5 in r such that the two vertices are endpoints in this path, by a 5 ;: 1. Hence we get a 2 :?: 1. But then the covering radius of C equals 2. By Theorem 3.20 this is impossible. O Lemma 3.24 The minimum distance of C is not 2.

.1 Introduction
We shaJl use standard terminology. A code C of length n over an aJphabet Q with q symbols is a. subset of Qn, i.e. we consider codes in a Hamming scheme. We denote the cardinaJity ICI of the code by M; dis the minimum (Hamming-)distance of the code. A l>all B,(c) with center c and radius ris defined by Suppose that Cis the union of two disjoint subcodes C 1 and C2 such that the following holds. There are integers r 1 a.nd 1· 2 sch tha.t the halls Br(c) where r r; if c E C; (i 1,2) are disjoint. We then call C, with the specified subcodes C 1 and C2, a (ri.r2)error-correcting code. Let M; := IC;I and let d; be the minimum distance of C;. Then d; ~ 2r; + 1 (i 1, 2). If we also define then it is also clea.r that Define r( c) = r; if c E C;. If c and c' are codewords with then c and c' will be caJled adjacent. In this way a graph is defined on the vertex set consisting of codewords of C. The code C is called bipartite if the two sets C1 and C2 are cocliques (independent sets) in this graph. If then C is called a perfect ( ri. r2)-error-correcting code. Let V ( n, r) denote the volume of a hall of radius r in Qn, i.e.  V(n,r)=V(n,r-l)+(;)(q 1)', and frorn well-known relations for binomial coefficients we find If C is a perfect ( r 1 , r2)-error-correcting code in Qn, then by (3.3) we have (3.7) Perfect codes with distinctive radii were introduced by CoHEN & MONTARON [29,78].
Tbey were also studied by GUNDLACH [56) (see also [54,55]). The exa.rnples we mention below were already found by COHEN & MONTARON (all of which were also found by the present authors, before Prof. J.H. van Lint ca.lled Gundlach's work to their a.ttention; no new exa.mples wen,-fouud). The a.im of this secticin is to prove a. non-existence result that does not occm· in [56]. This theorem sta.tes tha.t Exa.mple 4 (below) is unique (in a certa.in sen se). Example 1. Let C 1 be a.ny r-error-correcting code a.nd let C consists of the words of C 1 and all words in Q" with dista.nce > 1· to C1. Then C is a trivia.l example of a perfect code with distinct protective radii, na.mely an { r, 0)-error-correcting code.
Example 2. Let C be the bina.ry repetition code of length n, a.nd C 1 ::: {O}. If r 1 + r 2 + 1 n thcn this is a perfect ( ri, 1· 2 )-error-correcting code. is a perfect ( e, e -1 )-error-correcting code (since we have IC'I q-1 ICI). Exa.mples ca.n be ma.de using the known perfect codes.
Example 4. Let P be a code wit.h the pa.ra.meters of a. Prepaya.ta. code of length n 2 21 1. Let "Jt be the union of P a.nd a.Jl words in Z2 tlrnt have distance 3 to P. In [112] it is shown tha.t 'Jt is a. perfect 1-error-correcting code. Now take the ex- Then Cis a perfect (2, 1 )-error-correcting code. Note tha.t if P is the Prepa.ra.ta. code, then 'Jt is the Hamming code (cf. [6)).
We ca.n now state the ma.in theorem of this section. Theorem 3.36 IJ C is a bipartite perfect (r, 1)-error-correcting code with r ~ 2, then r = 2 and C belongs to the family of codes mentioned in Example 4.
From Lemma. 3.37 it follows that the pundured code c' ( delete the Jast symbol) is an ( r 1 , r 2 -1 )-error-correcting code So not only does ( 3.14) hold hut we a.lso have (3.12) If we multiply both sides of (3.12) by q, subtract from (3.7), and apply (3.5) and (3.6), the result follows. D Remark. Since (3.10) and (3.11) ca.nnot hold simultaneously unless q = 2, we have a second proof of Lemma 3.38.
Proof. The first a.ssertion is a direct consequence of Lemma 3.40. To prove the second assertion, assume <1 1 2: 2r1 +a, d2 2: 2r2+2. Now Cis a.lso a.( r1+1, r 2 -1 )-error-correcting code. So we have (as in (forin7) Proof. If (3.10) and (3.11) both hold, i.e. (3.13) holds, then we must have equality in (3.12), showing that. C 1 is perfect for (r 1 ,r 2 1). The second assertion follows in the sa.me way. O There is a pa.rt.ial converse to Lemma. 3.42. We state it, although we shall not need it Ia.ter.

Proof of Theorem 3.36
We remind the reader of the Johnson bouml for bina.ry codes (cf. (73]). If C is a binary e-error-correcting code of length n, then If equa.lity holds in (3.17), then either e + 1 divides n + 1 in which case C is a perfect code, or otherwise C is called nearly perfect. Note that if the fra.ction on the left-hand side of (3.17) is not 0, then it is at least H:). Therefore (3.18) implies that e + 1 divides n a.nd that C is a nearly perfect code. Now a.ssume that Cis a bipartite perfect (r, 1)-error-correcting code with r ~ 2. Apply Lemmas 3.41 a.nd 3.42. We find that C 1 is a perfect (r,0)-error-correcting and that It follows that C~ is an r-error-correcting code with equality in (3.18). Therefore c; is nearly perfect. It is well known (cf. [72]) that ( r ;::: 2) this implies that c~ has the para.meters of a Preparata code. This completes the proof. O

A characterisation of the Doob graphs
For a graph r we say that À exists if every edge lies in exactly À triangles. EGAWA [43) has shown that the Hamming graphs are determined by their parameters, unless À equals 2. In tha.t case the only possible graphs are the Doob graphs. RIFÀ & HUGUET [90] and NoMUllA [85] generalised this result by Egawa for À:/; 2. We generalise Egawa's result for À= 2.
The main part of the proof is a chara.cterisation of cartesian products of graphs. This chapter is based on the preprint "A chara.cterisation of the Doob graphs" [69).
In this pa.per we look at the case À 2. We shall prove the following result.  Using a result of HoNG [61] we get the following result as a corollary of the >. #-2. The paper is organised as follows. In Section 2 we give the definitions and notations. In Section 3 we discuss when a graph is covered by a cartesian product of gra.phs.
In Section 4 we look at the local structure of a graph with c2 = 2, c3 = 3, >. = 2 and a 2 4. In Section 5 we give the proofs of the above results.

Definitions and notation
For two graphs r, il, a graph homomorphism ' ljJ is a map i/l : A partition Pof the vertex set of a graph f into (non-empty sets) is called regular, or cquitable if, for any two C',C E P, the number of vertices y E C .adjacent to x E C' is a constant e(C',C) independent of :i: E C'. A regular partition Pis uniformly regular if tbere a.re çonstants e 01 and e 11 sneb tha.t A regular pa.rtition P is eomplctely regu.lar if all C E P a.re completely regular with the sa.me pa.ra.meters. Note tha.t. a. completely regular partition is uniformly regula.r.
A subgraph ll of a gra.ph r is i-conve~: if for all x, y at dista.nce j :5 i, the pa.tbs between x and y of length j in r a.re a.lso in ll. It is i-superconvex if, for all x, y at distance j :5 i, a.ll paths betwe(~n x and y of length at most j + 1 in r are in Ll. An i-convex The weight w( u) of a. vedor u is the number of i such tha.t u; #-0. If u and v are twó vectors of the sa.me length a.nd A is a set. of coordinates, then u IA v if and only if u; = v;  [41].
A Doob graph is a gra.ph isomorphic to D( c, s) for some integers c and s. We consider the Hamming graph H(n,4) also as Doob graphs.
We now give some examples of some of the above definitions.
(i) Each maximal clique in a cartesian product of cliques is 2-closed.
(ii) The Shrikhande graph has no proper 2-dosed subgraphs containing an edge.

On products of graphs
In this section we look at when a graph is covered by a cartesia.n product of gra.phs.  In the rest of the proof we write II for TIF;. In the followiug we wil! define the graph homomorphism 1r : Il -> r. We will show that ir will sa.tisfy the following two conditions.

Conditions (I) and (Il)
Let î.ï I; u, û 1 1 u and z 1 1 "Jl u with 'ü; = z; = a; and Ûj = Zj = ai. By ( ii) the vertices 7r(î.ï) and 7r(û) have exactly common neighbours. Define 11'(u) as the common neighbour of 7r(u) and 7r(û) different from 7r(z). We first show that 7r restricted to the set of vectors of weight 2 is well-defined. By ( ii) we  couta.ining x, the vertex y is not a. vertex of il>;;. It follows that d(yk, Zk) = 3, by the follows that z3 "" t, and the subgra.

On the local structure
Let r be a connected gra.ph with tt = 2, c 3 = 3, À = 2 and a 2 = 4. In this section we look at the local structure of r. First we give two lemmas. Proof. The vertices x; and y have two common neighbours in f(y) n f 2 (x), for i = 1, 2.
But jf(y) n f2(x)j = 3, and hence x1 and x2 have a common neighbour in f2(x) n f(y). Also x 1 and x 2 have exactly two common neighbours. D   Proof. Suppose n ;::: 4. By previous lemma we get n ;::: 6. If n ;::: 7, then there is a j such that d(yi,z) 3 So we have shown that z"' x 3 or z"' x 4 • But now byµ = 2 it easily follows that x, y, z lie in a 4-clique a 1 such that for each verte~ u of a 1 there is exactly one i such that u"' x;. D As an immediate corollary from Lemma 4.12 and 4.13 we have the following lemma.
Lemma 4.14 The configuration is not an induced subgraph of r. 0 Let X1 be a vertex of r and suppose that X1 is in a Shrikhaude subgraph, say a. Let Proof. Suppose that d(x;,x;) = 2. Let Xk be a common neighbour of x; and Xj· There is a vertex u 2 of Ilk, which is a neighbour of u 1 and also a vertex u 3 of II; such tha.t u2 ....., u3. By Lemma 4.16, u = tt3. Now by Lemma 4.14 we get a. rontradiction. 0  If C(f) is a. 4-dique, then follows with the same reasoning that x, y, z lies in a direct product of two 4-cliques. D

The cbaracterisation of Doob graphs
In this section we prove the results we mentioned in the introductiou. Remark. A related characterisation of the cartesian product of cliques is given by MOLLARD [77], Theorem 1.
NoMURA [84] has shown that connected graphs with c2 = 2, c 3 = 3, a 2 = 2À-:/; 4 do not have induced K 2 , 1 , 1 • The following theorem of NoMURA [85) and RIFÀ & HUGUET [90] is now a direct consequence of the previous theorem. By Theorem 4.4 we are done. D Before We sta.te the next theorem we first define quotient graphs. Let r be a. graph and let P be a partition of V r. Then the qti.otient gmph r /P is the graph with vertices the classes of P and CD is a.n edge if <lr( C, D) = 1.

Corollary 4.24 The partition P
{ Cx 1 x vertex of r} of the Doob graph ó., where C" = 1r-1 ( x) is a tmiformly regu.la.r partition of ó. and r is isomorph to ó./P. IJ r is distance-regular, then P is a completely regular partition.
It is obvious that II is a uniformly regular pa.rtition. The gra.ph ó. is a distance-regula.r gra.ph and therefore by [19), Let G be the perfect terna.ry Golay code generated by the rows of the circulant ( -+ + + + --+ -)n. Then G is a ternary [11,6,5] code. Let r be the coset graph of G, that is, the graph witll as vertices the 3 5 cosets of G in F~ 1 , w here two cosets a.re a.dja.cent when their difference contains a vector of weight one. Then r is a strongly regular graph with parameters ( v, k, À,µ) = (243, 22, 1, 2), known as the Berlekamp-va.n Lint-Seidel graph.

47
The automorphism group of Ll bas shape 3 5 : (2 x M 10 ), and acts edge-transitively with point stabilizer isomorphic to M 10 . The orbit diagram is v = 486.

Structure of the group; related graphs
In order to describe the group of automorphisms more precisely, we have to specify the representation of2 x M 10 inside GL(5,3 (2 x Mw).
The graph E can be constructed as follows:

Uniformly geodetic graphs
In this we give two constructions for uniformly geodetic graphs. Also we give sufficient conditions to assure that the graphs constructed are uniformly geodetic.

Construction 1
In this subsection we give a construction for uniformly geodetic graphs. Let distance l-or-2 graph f1 U f2 of a graph r bas Vf as vertex set and edges xy if dr(x, y) E {1, 2}.

Construction 2
We give now a second construction, which was first described in BROUWER & KooLEN [22]. Let r be a graph. Proof. Let 6 be the distance-2 graph of r. It follows from Lemma 5.3 that 6 '=! J (2k 1,k-1). Is x is a vertex of r, then we denote by x the corresponding vertex of 6. We construct now the graph II from 6 as follows. The vertices of II are the vertices of 6 and the maximal k-cliques of 6 and edges of II have the form {x, C} where C is a maximal k-clique of 6 and x a vertex of il such that x E C. Note that II is the doubled Odd graph with valency k. If x, y are two adjacent vertices of Ll then there is a unique maximal k-clique C in 6 such that x, y E C. Define now the function </>:VII--+ Vf by  (iii) An Odd gmph 0,,. with odd valency.
(iv) A folded (4m + 1)-cube. Proof. We use a prime to distinguish the parameters of ~ from those of r. We shall first show that for all j ::; d the numbers~ai exist and that for all i, 2i + 1 ::; d we have On the other hand we have 1 C2A:C2k+l It follows that C2k+i = 1 + kp + a.zk(x, y) and thus a2k exists and c21:+1 satisfies Equation . Now we showed that <t2k+i exists and a2k+1 + a2k = >..
The only bipartite distal\ce-regular graph with these c; are the doubled Odd graphs, which are the graphs J(2t + 1,t,t + 1). CASE 3. r is a bipartite graph with µ = 2. For the numbers c; we have c; = i. The only bipartite distance-regular graph with these c; are the hypercubes. CASE 4. r is a. non-bipa.rtite graph with µ = 1. By (5.5) we obtain that d is even and a; = 0 for i < d and a.i :f:. O. The numbers c; = ril· By Theorem 5.5, the only distance-regular graphs with these parameters are the Odd graphs with odd valency. CASE 5. r is a non-bipartite graph with µ = 2. Than d is even and a; 0 for i < d, a.i :f:. 0, and c; = j for all j. By Theorem 5.6, the only distance-regular graphs with these parameters are the folded ( 4t + 1 )-cubes. CJ

Remarks.
(i} For r ~ J(2t-1, t-1, t) the graph B(f) was constructed by BROUWER AND KooLEN [22]. The gra.ph (B(f) is regular with valency t 2 + t + 1. The full a.utomorphisrn group of B(f) is isomorphic to D 8 x Sym(2t -1) acting transitively on the vertices, with vertex stabiliser 2x Sym(t -l)x Sym(t). The numbers c;(B(f)) = i 2 and hence the Johnson graphs are not the only uniformly geodetic regula.r graphs with c; = i 2 • More information a.bout these graphs is in [22].

Chapter 6
The Metric Hierarchy In this chapter we will consider the rnetric hierarchy for graphs, recalling work by J.B. Kelly, Deza, Assaoud, Terwilliger and others. In the second section we wil! define lattices and Euclidean representa.tions of graphs. Also in this section we will define root graphs. In the third section we will classify the amply regular root graphs. (We closely follow [19, § §3.14-3.15]; however, since there are several rnistakes in those sections we thought it appropriate to completely redo this piece of theory.) This culminates in Theorems 6.30, 6.40 and 6.42, where the case µ 2: 2 is cornpletely settled. In the last section we will classify the distance-regular graphs whose distance matrix has exactly one positive eigenvalue. This section is joint work with S.V Shpectorov, [70].

Introduction
In this section we will give an introduction to the metric theory. First we wil! give sorne definitions.    ( iii) The distance-matrix D(f) has exactly one positive eigenvalu.e.
SCHOEN BERG [95] gives the following characterisa.tion of metric spa.ces of negative type.

Theorem 6.5 (SCHOENBERG [95]) A metric space (X,d) is of negative type ij and only ij (X, v'd) is isometrically embeddable in a Euclidean space. D
Before we can formulate the following theorem we defiue a hole in a. la.ttice. A hole in a. la.ttice l Ç Rn is a. point of Rn whose dista.nce to the lattice is a local maximum. Let B(x) be the set of elements of Lat minima.! distance from x. The following two theorems give two characterisations of hypermetric spa.ces: Theorem 6.6 (AssOUAD [3]) Let (X,d) be a finite metric space. Then the space (X,d) is hypermetric ij and only ij there is a lattice L with a hole x and a map</> : X -+ B(x), such that d(x,y) !lief>(x)-</> (11)  We can use the last theor<>m to show that the Gosset graph is hypermetric.

L 1 Lo cal characterisations
In this subsection we give loca.J information about graphs in the metric hierarchy.
The most obvious consequence of the penta.gonal inequaJity is that the subgraphs 58 and ·~.
a.re forbidden ( where the last graph is forbidden only when d( a, b} ::;: 2) • in deed if we give the left vertices weight -1 and the right vertices weight + 1 we see that thè pentagonal inequality is violated. This gives a cha.racterisation for graphs with diameter two satisfying the pentagonal inequality: Proof. The weight functions -1, -2, 1, 1, 1 a.nd -2, -2, 1, 1, 2 and -3, -2, 2, 2, 1 (in order from top to bottom, first left then right} show that the hexagonal inequality forbids the first graph, the 8-gonal inequality also the third, a.nd the 10-gona.l inequality also the fourth. D Proposition 6.12 Let r be a connected amply regular graph. IJ r has two non-adjacent vertices x and z, lying in a induced quadrangle and does not contain the first, third and the fourth forbidden graph, then a2(x, z)::;: 2(À µ + 2). Proof. Suppose Q = (x,y,z,w) is an induced'quadrangle of r (i.e. x "'y"' z N w"' x, x 1' z, y 1' w }. No vertex is adjacent to precisely 3 vertices of Q. ( otherwise we would see the third forbidden graph ), or to two opposite vertices of Q ( otherwise we would see the first forbidden graph). Hence the neighbours of x are y and w, the vertices adjacent to all 4 points of Q, those adjacent only to x and w, those adjacent only to x and y, and those adjacent only to x. The common neighbours of x and z are y, wand those adjacent to all points of Q. The common neighbours of x and y are those adjacent to only x and y, and those adjacent to all points of Q. The vertices only adjacent to x have distance have either µ -1 orµ-2 common neighbours in f(x).
Proof. (i) By the first and third forbidden subgraph this follows ea.sily.
( ii) follows directly from ( i) and the non-occurrence of the first forbidden graph.
(iii) Among theµ common neighbours of y and z, one is x and at most one is in r 2 (x). The remaining neighbours must he in f(x).
(iv) follows directly from Proposition 6.12. D

The metric hierarchy for strongly regular graphs
In this subsection we study the metric hierarchy for strongly regular gra.phs.
The suspension of a graph ris the complete union r  (iii) v = 2(k + 2) S 27 and r is a subgraph of the Schläfli graph.
(iv) v = j(k + 2) S 16 and r is a subgraph of the Clebsch gmph.

Lattices and graph representations
In the first we define lattices and give some properties we need later in this chapter. For more information on la.ttices, see for example CONWAY AND SLOANE [31] or EBELING [42].

Lattices
In thîs sectîon we define the squared norm of a vector x = ( xi.". , Xn) in Rn to he ( x, x) = E~1 x~. A lattice in R n is a discrete set of vectors in R" which is closed under a.ddîtîon and suhtra.ction (and hence under scalar multiplication by integers). A lattice L is genera.ted by a set X of vectors if L = { L a"x 1 a" E z} . xEX We shall only look at la.ttices which are generated by a fiuite set. The dimension of a lattice L, denoted by dim L, defined by the dirnension of the vector space spanned by the elernents of L. An integral lattice is a lattice in which the inner product of any two vectors is integral. An integral lattice is called even if it contains only vectors of even squared norm. Note that a.n integral lattice is even when it is generated by a set X of vectors of even squared norm. The direct sum surn of two la.ttices L 1 and L 2 is defined if Li and L2 are orthogonal, i.e., if (xi,x2) = 0 for all x1 E Li, x2 E L2, as 61 A lattice L is called irreducible, if L L 1 (B L 2 implies L 1 {O} or L 2 {O}. Otherwise the lattice is called reducible. The vectors of of squared nonn 2 in L are called the roots of L. A root lattice is an integral lattice L in Rn generated by a set of vectors of squared norm 2. There are connections between root systems and root lattices, for more information see for example [19, Chapter 3] and [42]. W!TT [110] has determined all irreducible root lattices: Theorem 6.18 The only irreducible root lattices, up to isomorphism, are An (n ~ 1), Dn, (n ~ 4) and E5,E1,Es. D Remark. If a root lattice Lis the direct sum of the lattices L 1 and L 2 , then L 1 and L 2 are also root lattices.
We now describe the irreducible root lattices which arise in the above theorem. We write e; for the unit vector in Rn ha.ving an one in the i-th coordinate and zeros elsewhere.

•=l
The roots of Dn are the 2n(n -1) vectors ±e; ± ej, i < j. Note that our first example A" is represented on a hyperplane in R"+l rather than in R" in order to get a nicer representation, and in order to make the indusion An C Dn+1 ohvious. ( v) Defint> Et; {x E Es 1 i:1 + · · · + X6 = x7 + xs = O}. E 6 contains 72 :l2 + 40 roots.

Graph representations and lattices
In this subsection we mean hy 'squared distance' the squared Euclidean distance in Rn, i.e. the innt>r product of t.lw difförence with itself, and 'distance' means graph distance.
A (Euclidc<m) ffJ1ff.~mtatio11 of a graph r is a map p: Vf -. R" such tbat the images 62 of adjacent x, y E Vf have constant squared distance. We can consider an undirected graph also as a directed graph in which both arcs exist between vertices x and y if xy is an edge. Remark that if a graph has a representation, then in fact we are labeling the arcs of f with constànt norm vectors in such a way that the sum of the labels in a directed cycle (i.e. a connected graph in which every vertex has one incoming are and one outgoing are) add to O, the all-zero vector.
We usually write x for the image p(x) of a vertex x under p, and [x,y]:=(x y,x-y) for the squared distance of the images of x, y E Vf. The representation is faithful when p is injective. where the representation of~ is the representation of r restricted to ll.

Root representaions and root graphs
A root representation of a graph is a representation such that squared distances [x, y] = A connected graph having a root representation is called a. root graph. The na.me reflects the fact that the lattice L(f) of a root representation is a. root lattice: Proposition 6.20 Let r be a root graph. Then L(r) is a root lattice.
Proof. Since r is connected, L(r) is generated by the norm 2 vectors x 'fi (x, y E Vr, x '"" y ), and therefore is a root lattice. D Recall from Chapter 4 that a subgraph A of a graph r is 2-closed if for any two vertices in A their common neighbours in r are a.lso in A.  ( iii) The hafoed eube A( n) is t.he code graph of the binary code consisting of all words with even weight (i.e. the number of positions with a one is even). It has 2n-l vertices, valency (;) andµ 6. A(5) is known as the Clebsch graph.
( iv) The Johnson graph J( n, t) is the code graph of the binary code consisting of all words oflength n and weight t. J( n, t) has G) vertices, valency t( n -t) and µ = 4. In particular (t = 1 ), cliques of size n are code graphs.
( v) The graphs B( n, t). Say that a binary vector x = ( Xi, x 2 " .• , Xn+2) has weight ( W1, w2) where W1 = x1 + x2 and w2 = X3 + X4 + ... + Xn+2• where we add in Z. Now B( n, t) is the code graph of the binary code of all words of length n + 2 with weights (2, t), (1, t -1), and (0, t) and has µ = 4. An B( n, t) is regular if and only if n 2t 1. The graphs B(2t 1, t) are first constructed by BROUWER AND KOOLEN [22). In that paper you can find more information about these graphs. In Chapter 5 we construct the graphs B( n, t) with a more generaJ construction.
( vi) The icosohedron with 12 vertices, valency 5 and Jt 2 can be viewed as the code graph of the binary code consisting of the words 000000, 110000, 0011111, 111111 and those obtained by a cyclic permutation of the first five entries. ( ix) The Petersen graph with 10 vertices, valency 3 and µ = 1 can be viewed as the code graph of the binary code consist.ing of the words 000000, 100100, 001111 and those obta.ined by a cyclic permutation of the six entries.
( x) A polygon: a g-gon ca.n be viewed as the code graph of the bina.ry code consisting of the word of length g with on the first l~J entries ones and zeroes on the other entries, and those ohtained by a cyclic permutation of the g entries.
(xi) If r and .'.l are root graphs, then the cartesia.n product II r x .'.l is aga.in a root graph: simply represent ( x, y) by the sum of the images of x and y. In particular, Hamming grophs (cartesian product of cliques) and Doob gmpl~~ (ca.rtesian product of 4-cliques and Shrikha.nde gra.phs) are root gra.phs.
We first prove some easy results on the locaJ structure of root gra.phs.  ( x, y ). This implies the first statement. The second statement follows by putting x y. o A direct consequence of Theorern 6.23 and Proposition 6.13 (note that graphs of negative type satisfy the 10-gonal inequality) is the following proposition. Proposition 6.25 Let r 'be a root graph. Then the /olfowing holds:

Proof. lt is straightforward that
(i) IJ x E vr, then two nonadjacent neighbours y,z of x have at most one common neighbour in fz(x) and such a neighbour has label y + z -~.  ( iv) IJ r is an amply regular graph wit/1 µ(r) = µ and À(r) = À, then for any two nonadjacent vertices x' y' lying in an induced quadrangle of r' we have a2( x; y) = 2( À -µ + 2).
Proof. (i) Let u be a common neighbour of y and z at dista.nce 2 frorn x. Then

Classification of root graphs
Using the classifica.tion of root lattices we sha.ll give a complete cha.racterisation of distance-regula.r graphs with µ ~ 2 la.ter in this section. In addition in Theorem 6.40, we give a complete cha.racterisa.tion of amply regula.r root gra.phs with µ ~ 2, by use of computer result.s.
(y, y) does not depend on y, such tha.t [x, y] is an even integer for a.ll x, y-E Vf and [x, yJ = 2 if and only if x "' y.
A gra.ph f has an integml mot 1"epresentation if it ha.s· a. root representa.tion such tha.t L(r) Ç zn for somt> n, i.e. the label are elements of zn. First we will consider root gra.phs with a.n int.egral root representa.tion.

Integral root representations
In this subsection we classify the amply regular root gra.phs with a.n integra.1 root representa.tion.  Proof. Without loss of genera.lity wc may assume that x = 0 and y 2e1. Lemma 6.24 shows that common neighbours of i: and y are represented by suitable vectors e1 ± e1, and, by pcrmuting the e1 and cha.nging their signs if necessa.ry, we may a.ssume that M := f( x) n f(y) is ret>rese1tt.ed by the 11 = 2s + t vectors CJ ± C/ ( l = 2, ... , S + 1 ), CJ + C/ ( l = 8 + 2, ... , S + t + 1 ). We may assume that x e; + ej a.nd xy ek + ei, with distinct i, j, k, l. Proof. The first claim follows immediately from Proposition 6.25 (i). If µ is constant, then the representation of { x} U f( x) determines the structure and representation of r 2 ( x ): for any two non adjacent vertices u, v E f( x) that have only 11 -2 common neighbours in f(x), we find a vertex y represented by 'ü + v -x. But the restriction of the representation to r <2( x) is injective, and determines the structure of this subgraph, by Lemma 6.28 above.-Since root graphs are connected, we are done. D Before we ca.n state the next theorem we first have to define the gra.phs L(s, t).
An L( s, t) is the graph with vertex set V = S U T, ISI = 3s + t, ITI = t. The subgraph induced by S is a clique. A vertex t E T is adjacent to exactly 3 vertices in S a.nd the distance between any two t 1 , tz E T is three.
( iii) µ, = 5 and r is the suspension of a triangular graph. ( iv) µ, = 4 and either r is a Johnson graph or r ~ B( n, t) for some n > t ~ 1.
( v) µ, = 3 and either t is the suspension of the (p x q )-grid, r is an L( s, t), or Furthermore, _all the above graphs have an integral root representations.
Proof. By Proposition 6.25 (iii), I' is locally connected. Suppose that r contains vertices x, y at distance 2 such that the norm 4 vector yx ha.s the form 2e j for some j. Since r is loca.lly connected, it follows from Lemma 6.26 that f( x) n I'(y) = I'( x ). First a.ssume that f2{x) i-{y} and let z E f2(x), distinct from y. Then f(x) = f(x) n r(z), since r( x) has cardinality /t. So we may a.ssume without loss of genera.lity that x O and z = ei+ e, +es+ e 1 , for some r, s, t. The vertices of r( x) a.re represented by ej + e1, with l E {r, s, t}. So /t = 3 and fis K3,1,1,1· If f2(x) consists only of y, then it is easy to see that we are in case (i).
Fix a vertex x of r. By Proposition 6.27 we may assume that 'il L; y;e; with Yi E {O, 1} for all i and y E Vf, and that x = 0. Now let ~ be the graph whose vertices are the n indices 1, ... , n, and whose edges are the pairs ij such that some point off( x) is represented bye;+ ei. Since distinct vertices of f( x) are represented by dis tin et vee tors ( their difference must have norm 2 or 4) we ma.y identify the point represented by e; + ej with the edge ij of Ll.
Let US denote by S(y) the set of indices i with Y; = 1 for y E vr. By Lemma 6.24, the vertices y E r 2 ( x) are represented by norm 4 vectors 'il = 'fi -x, and these must have the form e; + ej + ek + e1, i.e., S(y) is a 4-set. The µ common neighbours of x and y are certain edges of the subgraph of Ll induced on S(y). By Lemma. 6.28, the intersection f(x) n r(y) coincides with the set of edges of~ contained in S(y), and y E f2(x) is uniquely determined by S(y). To simplify the notation we shall use the abbreviation 0:/31/i fora 4-set {a,/3,1,6}.
Let us call a set. S Ç { 1, 2, ... , n} special if there is a vertex y E r with
Sl. If y E f 2 (x), tlwn S(y) is special.
(Indeed, if afJ1é and ée represent vertices at distance two, then they have µ common neighhours, and these must he among the six sets listed. On the other hand, if one of these six sets is special, we must show that it represents a common neighbour of afJ1é and&. But this follows immediately from Lemma 6.28.) By definition, the edges of ó. are just the special 2-sets. 8ince w;:::: 3, 83 implies that ó.
bas no subgraph consisting of two disjoint edges. Therefore ó. consists of a connected graph ó. 0 and a (possihly empty) set of isolated vertices. 83 implies S5. Any two disjoint edges of ó. have µ 1 or µ -2 transversals (i.e. edges meeting both given edges).
(iii) ~o is either complete bipartite or complete bipartite 'With one additional edge. lndeed, if A contains two dis joint edges, then ( since these must have at least two transversals) A contains a triangle or a quadrangle, and since B :j:. 0 this is forbidden by (ii). Thus, all edges in A pass through one vertex 6. lf both A and B contain an edge then ~o contains a K 4 • So we may assume that Bis a coclique. If the cardinality of B is at least 2 and A contains two intersecting edges, then ~o contains a K 1 , 2 , 2 , contradiction. So we may assume that B = {{J}. If A has no isolated point, then we can move ó from A to Band are done. So we may assume that A bas at least two edges 16 and 6e and an isolated point a. Now S4 yields a contradiction.
(iv) lf ~o is complete bipartite; say K,, 1 , then r ~ J(s + t, t). lndeed, this follows from Proposition 6.29. Now assume that ~o is a complete bipartite graph Ks,t with one additional edge in the t-coclique, so that f( :t) is a grid s x t with one additional vertex adjacent to a s x 2 subgrid. Let us temporarily ca.11 this Jatter graph s+ X t -in the figure below the graph 4+ x 5 is dra.wn; a label ij denotes the vector tt = e; +ei.
If E "' f3 and E "' i, then S4 with ({J, ó) interchanged gives a contrádiction. Therefore E is adjacent to just one of /3 and 7 (and if E "' fJ then E{37ó is special). Hence the neighbours distinct from 7 of fJ form a codique A, and the neighbours distinct frorn f3 of 7 form a coclique D. Now it is easy to see that do AU {/3, 7} UD (the coclique extension of P where a and ó are blown up to A and D, respectively). The set a{J7ó is special for alla E A and ó E D. Let a E A, and /3e E D. Then by S4 with (î',ó) interchanged we find that 7óe17 is not special for all 17 E D. It follows that there are no more special 4-sets. With the same a.rgument and Proposition 6.29 we see that there are no special 6-sets or non-special 4-sets at distance 3. We obtain that r is the suspension of the ((IAI + 1) x (IDI+ I ))-grid. It is easy to see that the cartesian product ö 1 x A2 is a subgraph sneb that vertex ( u, v) E lf A1 X V D.2 has label ü + v, since the la.helling of 6 1 is injective and x1 is ortbogonal to X2 for x; E V D.,. Now it is obvious that this subgraph is a connected 2-closed subgraph of r. D Proposition 6.32 Let r be a root graph with Jt(f) = 2. Let C be a clique in f. IJ C is a 2-closed subgraph of r then r is the cartesian product c x D., where,:}. is a 2-closed subgraph of r.
Proof. Let x E C and shift the representation such tha.t x O.
Let y E f(x) \ C.
Then, by the 2-closedness of C, y is perpendicular to ë for all c E C. There is a set T of neighbours of y such that for ea.ch c E C \ { x}, there is a t E T with t = y + ë. The subgraph induced by T U {y} is a 2-dosed clique, otherwise with the inversal procedure we get that G is not 2-closed. Therefore l -y is perpendicular to vy for all v E f(y) \ T and all t ET.
Le\ II be the graph with vertex set Vf and uv is an edge in II if uv is an edge in r and ü -vis orthogonal to ë for all c E C. Let 6 be the component which contains x. With induction to the distance of two vertices in 6 it is easy to see that 6 is a 2-closed subgraph. By the previous lemma r has an subgraph isomorphic to the cartesian product C x 6. But this must be r itself. o Before we can state the following proposition we first need to define the graph L.
The vertices of L are the 192 subsets A of the vertex set of a Petersen graph II such that the induced subgraph on A in II has one of the following types, and A "' B if a.nd only if IA+BI = 2: 0,

. '
and The graph L is locally the line graph of Petersen and with respect to the above partition of the vertex set of L we have the following intersection diagram for L: Reinark. The graph Lis an antipodal 2-cover of a graph 6. This graph 6 is a distanceregular graph with intersection array {15, 10, 1; 1,2, 15} and is locally the line graph of Petersen. See [23], for more information on L and rela.ted structures. Proposition 6.33 Let r be an amply regular root graph with /t(r) = 2. IJ L(r) ç zn, then one of the following holds.
( i) r is a Hamming graph, ( ii) À = 2 and r is a direct product of 4-cliques, icosahedra and Shrikhande graphs, and ( iii) À 4 and r is the direct product of 6-cliques and some copies of L.
The graph r does not contain vertices x, y at distance 2 such that the norm 4 vector x -'jj has shape ( ±2)on-1 . In deed, then Lemma 6.26 would show tha( r( x) has a component of size at most 2 and this contra.diets À ;::: 2. As a consequence, for any two vertices x, y at distance two, the vector x y must ha.ve shape ( ± 1 ) 4 on-4 • We can proceed as the proof of Theorem 6.30. By Proposition 6.27, we fix a vertex x and shift the representation such that x = 0 and all vertices of rare represented by E a;e;, where a; E {O, l}. Let p be the map which sends x to x. We define the graph A and the special sets as before and find that 81-85 remain va.lid; in particular, special 4-sets conta.in precisely two edges, and the union of two disjoint edges without a transversal is special. Let xy be an edge of r. We will show that the minimal 2-closed subgraph, sa.y II, of r containing the edge xy is isomorphic to either a (À+ 2)-clique, the Shrikhande gra.ph, the icosahedron or the graph L.
Let A 0 be the component of A containing a, where y = e" + e/3. Since f(x) is regular of valency À, every edge of Ao intersects precisely À other edges of Ao. If Ao contains a triangle, theu S5 implies that Ao is a triangle, and thus À = 2. This means that xy lies in a 4-dique. By Proposition 6.32 we may assume tha.t Li 0 does not contain a. triangle. Also Ao does not contain a. quadraugle, by S5. We need some further fa.cts. Applying 84 to a path of length 4 in A 0 we fiud: S6. If;,...., fJ '""e '""f3 "' a is an induced path in Ao, theu a;fJe is special. Indeed, S6 shows that 1345 and 1346 are special, so that S4 applied to 1345 and 46 gives a contradiction. We now distinguish three cases. CASE 1. Ao is a tree. Since L(Ao) must be regular of valeucy À, we fiud Ao !:! Kn,1• But then the edge xy lies in a (À+ 2)-dique. CASE 2. Ao has girth g at least 6. Then S8 implies that no vertex of a. g-gon ha.s a. further neighbour. 8ince A 0 is connected, it is a g-gon with vertices labelled in Z 9 and edges { i, i + 1}( i E Z 9 ), say. In particular, À 2. By 83 and S7, the sets { i, i + 1,j,j + 1} 6. The 16 special sets found so far represent the Shrikhande graph, a graph E with ..\( E) = 2 À,µ( E) = 2 and of diameter 2, this means that II is the Shrikhande graph.
CASE 3. Äo has girth 5. Then Äo is not bipartite, and since L(Ao) must be regular of valency À, Äo is regular of va.lency ! À+ 1. If À = 2, then Ào is the pentagon. Now it is clear that the edge xy lies in an induced subgraph isomorphic to the icosahedron.
lt is easy to see that this icosahedron is an isometrie subgraph of r and is a 2-closed subgraph. It follows that II is the icosahedron.
If À > 2, then for every vertex i of a pentagon 1 "' 2 "' 3 "' 4 "' 5 "' 1 of A 0 , the set Ao( i) of neighbours of i is a ( p -1 )-coclique, and by S7, the induced subgraph on with the special 0-,2and 4-sets we already found. By induction on the distance, it is easy to show that for all vertices z E VII there is a component in II(z) whose labels are subsets of V Ào. lt follows that Il must be locally the line graph of the Petersen graph. Also by induction on the distance you easily see that the la.helling of Il is unique. We a.lready saw that the graph Lisa. locally L(Petersen) root graph and therefore II must be L.
This cornpletes the proof of the proposition. D

Amply regular Terwilliger root graphs
In this subsection we classify the amply regular root graph without an induced quadrangle. A Terwilliger gmph is a non-complete graph f such that, for any two vertices 7, é at distance two, r('i') n f( é) is a clique of size p (for some fixed JL ;:: 0). In other words, a. Terwilliger graph is a non-complete graph r without· induced qua.drangles such that a.ny two nonadjacent vertices have 0 or 11 common neigbbours. They were studied by TER.WILLIGER [104]. f(x). Without loss of generality we may assume that Zi = e; + e;+3· Also without loss of generality we may assurne that y;,y;_ 1 E a(x;) for i E Zs. The label of y; must be integral by the labels of z; and Zi-1 · If Yi has an integral label then Yi = e;+e;+1 +e;H +t: where t: E {±e1, ±es}. But tben follows a contradiction by looking at y 1 and y 4 • D Lemma 6.37 Let r be the distance-regular gmph with inte1·section diagram v=65.

5
Then f is not a root gmph.
From this it is easy to see that the subgraph a induced on r 3 ( x) is the disjoint union of two icosahedra. Let G be the group fixing both icosahedra in r 3 ( x ). Then G ~ Alt(5) x 2.
It is ohvious that T n G ~Alt (5) and therefore there is an automorphism 1l' of Aut(f) fixing the set { x} U f( x) and' sends each u E f 3( x) to its antipodal in the component of A where u is a vertex of.
Suppose that r has a root represeuta.tion. The lattice L(f) contains E6 as sublattice, because this is the lattice belonging to the suspension of the Petersen graph. Also L(f) is irredudble by the locally connectedness, so L(r) Ç Es. pointwise we obtain f( x) n f 2(Y) = f( x) n f 2( z ). From the fact that at least three elements off( v) n f 2(x) must have au integral label where v E {y, 7r(y)}, there is a vertex u E f(y) n f 2 (x) such that both i1 and 7r(u) have an integral label. But this means that the sixth entry of 1r( u) must be -L Therefore (y -1r( u ), y -1r( u)) > 6, what is impossible, because they lay in a.n icosahedron. D The condusion is: The only amply regular root gmph without induced quadmngles is the icosahedron. D

Classification of root graphs
In this subsection we dassify first the distance-regular root graphs. We a.lso classify the amply regular root graphs, using some computer results.
Distance-regular graphs Proof. Fix x a vertex of r, and shift the representation such that x = 0. Then L+(r(x)) Ç L(r) s E 8 , and the induced subgraph induced by f(x) has smallest eigenvalue -2, since Proposition 6.14. Looking in the list of the strongly regular root graphs we find that if d = 2 then r must be one of the Cha.ng graphs, or the Schläfü graph.
(Note that L(r) does not depend on the representation if the diameter is at most 2).
From now on we assurne tha.t d ~ 3. If r does not contain induced quadrangles, then by Proposition 6.38 we find that r must be the icosahedron. So We may assume tha.t r has an indured quadrangle and thus by Proposition 6.25 (iv), we have k = b 2 + 2À + 4 µ.
Also we may assurne tha.t there a.re vertices y, z, u such tha.t x "" y "' z "" u "" x and d(x,z) 2 = d(y,u).
By Proposition 6.32 we see tha.t r does not conta.in (A + 2)-cliques. By Proposition 6.25 (iii), r is locally connerted, except possibly when Jt 2. But then À ~ 2 and each component off( x) is a regular graph conta.ining a.n induced g-gon, with g ~ 5, because f( x) does not conta.in a.n induced qua.drangle, otherwise JI ~ 3. Rut then ea.ch component of f(J:) contrib11tes at least 5 to the dimension of t+(f(3:)) a.nd thus there can be only one component. in I'(l:). Therefore f(x) is connected regular graph with k vertices and va.lency, and k ?:: À+ Jt + l, by (19], Theorem 1.5:5. We now use Theorem 6.16 to determine the possibilities for f( x ). Note tha.t À ?:: 2 by locally counectedness.
Let Il the gra.ph indnced by 1'(3:) n f(y). Let t be the nuinber of edges in II.
We count the edges bet.ween f(.1:) n f(y) and f(x) n f 2 {y). Note tha.t any non-adjacent vertices in f(x) have at least JI· -2 common neighbours in f(x). STEP 3 The gmph n has minimal t1<1lcncy at least /t -4.
Let w be a vertex of Il. If there is no vertex at distance 2 in Il, then by>. ;::: µ -2, the valency of w is at least µ -3. Otherwise let a be a vertex of n at distance 2 from w. The vertices a, w, x lie in at most one induced quadrangle of r. The same holds for the vertices a, w, y. Now this step follows easily. (iv) µ = 5 and>. E {8, 12}.
For /t = 2 and >. 2, we get that r is locally the 8-gon, what is a line graph. If /t = 3 and À= 3 then l'(a:) lms vaJency 3 and each two non-adjacent vertices in f(x) must have at least one common neighbour in f( x ). Therefore r must be locally the Petersen graph, hut none of the locally Petersen gra.phs have Il = 3, see Theorem 1.3.
If ,\ + 3 < c 3 < 2.\+ 4, then we have the following possibilities for (µ, ..\, c3); ( i) (À,µ, c3) =  [19], and hence f(x) is a strongly regular root graph 6 wit.h v (6) From now on we suppose that f(x) is a line-graph. Hence we may assume that f(x) is the line graph of a graph ó. with n vertices. By Theorem 6.16, ó. is regular, or bipartite and semiregulax. Moreover, in the bipartite case L+(f(:r)) ~ An-l and hence n ::; 9, and in the other case L+(f(x)) ~ D.,. and hence n ::; 8. By Proposition 6.25 (iii), non-adjacent vertices of f( x) have µ l or µ -2 common neighbours in f( x ), i.e., disjoint edges of ó. have µ -1 or µ 2 transversals (i.e. edges intersecting the given edges). In particular Jt ::; 6. Using the fact that x lies in an induced quadrangle, it is a simple exercise (cf. Proposition 5 of NEUMAIER [82]) to show that we have one of the following cases.
(iv) µ = 4 and ris either a Johnson gmph, or one of the Chang gmphs.
( v) µ = 2 and either r is a Hamming gmph, a Doob gmph, i.e. a cartesian product of 4-cliques and Shrikhande graphs, or the icosahedron.
Proof. Suppose first that /t 2: 3. Then r is locally connected and by Proposition 6.22 L(f) is irreducible. If L(r) Ç Dn for some n then r has an integral root representation and by Theorem 6.30 we have cases ( iii) and ( iv ). If L(f) E; then by Theorem 6.39 we are in the the cases ( i), ( ii) or ( iv ).
So we may suppose that µ = 2. If r does not contain an induced qua.drangle then r is an Terwilliger amply root gra.ph and by Proposition 6.38 r must be the icosahedron. If L 1 Ç Dn for some n, then II 1 is a, Shrikhande gra.ph of a. (À + 2 )-clique. By Lemma 6.31 and the regularity of r, the gra.ph ris the cartesian product II 1 x Il2. Therefore r is a. Ha.mming graph or a Doob graph. D Amply-regular graphs BROUWER, COHEN & NEUMAIER, [19,Proposition 3.15.2] cla.ssified the amply-regular root graphs with L(r) e! E;, using a computer search by Bussemaker and the tables of regular graphs with smallest eigenvafoe -2, hut they forgot to mention the icosahedron. Proposition 6.41 Let r be an amply 1-egular mot gmph of diameter d with parameters (v,k,>.,µ.). /fL(f) ~ E6>E1,Es, then either we have d > 2, k S: 8 andµ= 1, or ris isomorphie to the icosahedmn, one of the Chang graphs, the Schläfli graph or the Gosset gmph. D Now we are ready for the cla.ssification of the amply regular root graphs with µ = 2.   Proof. If xo "" x1 "' ... "' x. is a geodetie path then by Lemma 6.44, for any 1 :5 i < j :5 s, the roots x;-x;_ 1 and x;-x;_ 1 are perpendieular. By assumption, x;-x;_ 1 ±e"±e 0 and Xj -Xj-1 = ±ec ± ed for disjoint {a,b} and {e,d}. It means that for all vertices x and y in r every coordinate of the vector x -y (in the base { e 1 , ••• , en}) is equal 0 or ±1. .It implies that up to a shift the vectors x E X belong to the natura! cube, and this, dearly, provides an isometrie embedding of r into the associated halved cube. D As we already observed, we may restrict ourselves to the case tt = 1. This implies, in particular, that every edge of r is contained in exactly one maxima! clique. Let g denote the geometrie girth of r, i.e., the minimal length of an induced cycle, other than a triangle.

The geometrie girth is even
In this subsection we look at the ca.se that the geometrie girth is even. Fora pa.th p define R(p) to be the set of all root vectors x y E L(f) corresponding to all edges { x, y} where at least one of x and y belongs to p. We claim that R(p) does not depend on p. By connectivity of E it s11ffices to check the claim for two adjacent paths. Let xo"' x1 "' ... "';r. and .1:1 "'x2"' ... "' Xs+i be such paths (call them P1 and P2).
Since this cycle is minimal, we know the distances between vertices on it. Applying Lemma 6.44, we esta.blish tha.t y -x = Xs+2 -Xs+I· Hence R(p1) Ç R(p2). By symmetry, it implies the equality. Now, for a pa.rticula.r path p of length s, the suhgraph induced by the set of vertices at dista.nce 1 from p is a. disjoint union of cliques. Lemma 6.44 easily implies that v, u E R(p) are perpendicula.r if a.nd only if the corresponding edges do not belong to the sa.me clique. It. follows that. L is a sum of la.ttices Am 's and hence L is a sublattices of An for a. sufficiently Ja.rg<' n. By Lemma 6.:16 we establish that f is an isometrie subgraph of a halwd cubC'. D Proposition 6.48 Let f /x· a dista11ce-1-egular graph with even geometrie girth g 2' . 6. IJ r satisfies the JK:ntngonnl inequality, tlien r is a do11ble<l Odd gmph or a polygon.
The graph fis bipartite, by step 3, and hence isometrica.Jly embeddable in a hypercube. So the halved graph is an isometrie distance-regular subgraplt of the halved cube with Jt ;::: 4. By Theorem 6.40 we get that the ha.lved graph of r is a Jolrnson graph or a halved cube. HEMMETER [58,59] found tha.t a Jolrnson gra.ph is only the halved graph of a doubled Od<l graph and a halved cube is the halved gra.ph of a. dista.nce-regular graph with Jt = 2.
This completes the proof of the proposition. D The conchtsion of this subsection is tha.t the distance-regnlar gra.phs with even geometrie girth and of negative type are exa.ctly the doubled Odd gra.phs and the even polygons.

The geometrie girth is odd
In this subsection we look at the case that the geometrie girth is odd. Lemma 6.49 Suppose g = 2s + 1 is odd. Suppose also that euery geocletic path of length s is contained in an induced cycle of length g. Then L(f) is frreducible.
Proof. It suffices to prove tha.t for every two adjacent edges the corresponding roots belong to the sa.me irredudble component. By assumption these two edges are contained in a cycle C of length g. By Lemma 6.44 and by the minima.lity of C, the roots corresponding to two edges of c·a.t distance s -1 (maxima!) from each other are not perpendicular. Clearly, it implies that all the roots corresponding to the edges of C belong to the sa.me irreducible component. D By this lemma, in case of odd geometrie girth, L(f) is 011!' of the la.ttices A", Dn or En.
Therefore, t > s. Let now C be a cycle x0 ,..., ••• "" x 8 "" Ys+i "' ... "' y 9 x 0 • Since Ys+l -x. is not perpendicular to ei -e2, it is not perpendicular to e1 + e2, either. Since    [45] has shown that there are no distance-regular graphs with intersection arrays ( ii) and ( 1•i). For the remaining intersection arra.ys one can find the .eigenvalues of the dista.nce matrix, and see that in ea.ch case there exists a second positive eigenvalue. CASE 2. Suppose À = 1 and r ?: 3. Then k 6 or 8.
The distance-regular graphs with valency 6 and À = 1 were cla.ssified by HIRAKI, No-MURA & SuzuKJ [60]. The only graph they found with odd geometrie girth is the graph with intersection array {6, 4, 2, 1; 1, 1, 4, 6}. It is easy to see that the distance matrix of thls graph bas more than one positive eigenvalue.

Standard representations
In Section 7.1 we discuss the existence of the standard representations of distance-regular graphs and develop some theory using ideas of Terwilliger and Godsil. Nain results of this section are an improvement of Godsil's diameter bound in Theorem 7.17 and a characterlsation of the cubes in Theorem 7.24. In the second section we will look a.t distance-regular graphs with ad 0 and discuss a new feasibility condition for distance-regular graphs with ad = 0. Using this condition we will show that as a consequence of Theorem 7.12 there do not exist distance-regular graphs for an infinite family of possible intersection arrays. This section is based on joint work with C.D. Godsil, [49] .. In the final Section 7.3 we will classify the distance-regular gra.phs with an eigenvalue with multiplicity 8 (Theorern 7.33). This is joint work with W.J. Martin.

General theory
We start by giving the expla.na.tion for the existence of standard representations.  In this chapter we will use x for the label of x in the standard representation corresponding toa given eigenvalue. If Ais set of vertices, we will use A instead of {iï 1 a E A}. Now we will show a lower bound on the multiplicity of an eigenvalue in a distanceregular graph.
A chordal graph is a graph such that between any two vertices there is a unique induced path connecting them. In Jl We just sa.w tha.t b"' c implies fJb = fJc-By substitution we obta.in: It follows that one simple nonzero fJb determines the others ( so dim W $ 1 ), unless I+(sb-l)u1-sbu2 = 0,i.e., sb = -bi/(8+1), (since u1 = 8/k and u2 = (8 2 -8>.-k)/(kb 1 )) and in this case dim W $ t, where t is the number of maximal (s + 1)-cliques in~ containing only x as cut vertex.
We have shown that if 8 = ~, then dim(W) $ max{l,t}, where t is the number of maxim al ( s + 1 )-cliques in ~ containing only x as cut vertex, and otherwise dim(W) $ 1.
So with the induction assumption we are done. 0 A corollary of this proposition is the following.  ( iii) Ass11me that eitlier r is not an antipodal cover, or r is an antipodal t-cover with t ~ 3 and 9 is not an eigenualue of the folded graph. If Cd-i ~ 2, then m > i + 2, and Using bm-1 = 1, GODSIL [46] showed that there are only finitely many coconnected distauce-regular graphs with an eigenvalue with a given multiplicity. In particular -l--->n>-1---(Jd + 1 -., -81 + 1 ' and equality in (7.2) can occur only for 9 = 9 1 or 9 (Jd· Moreover, equality in (7.2) occurs precisely when there is a vector z :/= 0 orthogonal to the all-one vector j with Gz O, where G denotes the Gmm matrix for V Li.
Proof. Let 8 be an eigenvalue of r different from k, and let u 0 , u 1 , u 2 , ••• , ttd be the standard sequence associated to 8. We denote the adjacency matrix of à by B. The Gram matrix G of V Li equals I + u 1 B + u 2 (J -I B) and all its eigenvalues are non-negative.
lt follows: Let q be an eigenva.Jue of B with eigenvector z orthogona.l to the all-one vector j. Then Gz = ((1-uz) + (u1 -uz)11)z, whence follows that we deduce inequality (7.2). This holds for all IJ# kof r and all eigenvalues 11 of ti. with an eigenvector orthogonal to the all-one vector j. In particular, inequality (7.3} follows by use of lid < -1 and IJ 1 ~ 0 and equality in (7.2) can occur only for 9 = IJ 1 and for Theorem 7.11 Let r be a distance-regular graph with eigenvalues k Oo > 6 1 > · · · > 6 d. Let A be an induced regular subgmph of r with valency l > 0 such that for all vertices x, y E VA the distance dr( x, y) is at most 2. Let fJ be an eigenvaltie· of r distinct from k and let x be the representation of x belonging to 9. Let w be the cardinality of VA.
is an eigemialue of A with multiplicity w dim(VA} -€, unless -1 --Î:h = l is the 11alency of A. In that case the number of components of A equals w dim(VA} -€ + 1. Proof. tet tto, Ut. ttz, ... , ttd be the standard sequencè associated to IJ. We denote the adjacency matrix of A by B. By assumption BjT = ljT. The Gram matrix G of VK equals I + u1 B + tt 2 ( J -I -B) and a.11 its eigenvalues are non-negative. By evaluating, we obtain that GjT = 0 is equivalent with t = 0, where j is the all-one vector. Let U denote the null space of G. By Proposition 7.10, equa.lity holds in (7.2) if and only if U contains a non-zero vector orthogonal to j and either fJ = 9 1 or fJ = (Jd· Let W be the subspace of U orthogonal to j. As U is an eigenspace of G and j an eigenvector we see that W = ll unless j E lf. In this last case we have dim(W) dim( U)-1. The condition jE U is equivalent with t = 0. So dim(W) = w -dim{VA) ê > 0. We find that W is an eigenspace of B with dgenvalue fj for which equality holds in (7.2 is an eigenvalue of the local graph r( x) with multiplicity at least k -m;. IJ In the next section we will see a generalisation of the above theorem. Now we will give an improvement of the diameter bound. First we need some lemmas and propositions. Using the theory developed by A.A. Ivanov [19,Chapter 14] we find that r must be the dodecahedron. o Lemma 7.15 Let r be a distance-regular graph with a 1 = a 2 = 0, a 3 ;:::: 1 andµ = 1.
Let 0 be an eigenvalue of multiplicity m ;::: 3 o/ r and let Il be an induced heptagon in r.
For k = 3 the zeroes are 0, l ± J2. It follows that 0 E Q(J2 Proof. We rnay suppose that d 2 2m 1 and hence ai = 0 by Theorem 7.9. Since bm-i = 1 (by Propositiou 7.8), we have Cm 1. Let g be the girth of r. Let t := L~J.
Since m 2 3 we have k 2 3. The distauce-regular graphs with valeucy 3 have been classified by BIGGS, BOSHIER. & SHAWE-TAYLOR [12] (cf. [19,Theorem 7.5.1]) and by checking them we see that. r must be the dodecahedron. lt has m = 3 and diameter 5. So we may assume that k ;::: 4. The distance-regular graphs with m = 3 are the 5 platonic solids, cf. GoDSIL [46]. The distance-regular graphs with m 4 have been classified by ZHU [114], cf. our future Theorem 7.30 (i), and all the graphs have diameter at most 6. So we may assume that m 2: 5. Suppose that t 2: 4. By Proposition 7.5 we have t S 2logk-i (m/2) + 1 and m ;::: k(k -1). Since k 2: 4 and by Proposition 7.13, the largest number i with c; 1, satisfies i S (kl)(t -1) + 1 S 2(k-1) logk-l (m/2) + 1 <m. So we fiud Cm 2 2, contradiction. Hence the girth of the graph is at most 7. Since Cm = 1 and m ;::: 5, the girth is 5 or 7.  We have m(l) 3, and m(2) = 4. Now we will look at the case that the valency k 2 2m. We first need the following lemma.

Lemma 7.19 IJ a is an algebmic integer, but not an integer, then /or all integers a,
there is an algebraic conjugate of a, say (3, with 1/3 -al > 1. Proof. The product TI(a;a) over all algebra.ic conjugates of°' is an integer not zero. So there must be at least one a; with la; -al ~ 1. So we are done. D Let r be a distance-regular graph with eigenvalues k = 8 0 > 8 1 > · · · > 8d with corresponding multiplicities 1 = m 0 , m 1 , ••• , md. Define b+ = _b_i_ and b-= ~. ft:ï is an integer. Now we have to consider two cases. First a.ssume that 8 = 8d. Then b-is an integer and by Theorem 4.4.3 of (19] it is a.t most -2. The loca.l graph ha.s À as eigenvaJue with multiplicity at least one and -b-1 with multiplicity at least k -m. So there are m -1 unknown eigenvalues, say 1ri, ... , 11'm-I • The trace of the adjacency matrix of the local gra.ph is 0, so the sum over all eigenvaJues is 0. We have to show that L: 1r; > -À( m -1) + 2. If the 1!'; are not all integral then there is at least one greater tha.n -À+ 2, by Lemma. 7.19, and we are done. So we may assume that the 11/s are all integral. If -.\ bas multiplicity e ;:::: 1 then À has multiplicity at least e and ,L1r;~-A+(m 2e)(--X+l)>-À(m-1)+2 (si nee m > 4 ). Otherwise L: 11"; ~ ( m -1 )(-À + 1 ), which is even la.rger.
Assume now tlmt 8 = 8 1 . Then b+ is an integer and by Theorem 4.4.3 of [19] it is at least 1. The local graph has À as eigenvalue with multiplicity at least one and -b+ 1 with multiplicity at least k m. So there are m -1 unknown eigenvalues, say 1!'1, ••• , 1r m-1 and we have to show that 1!'; < À( m 1) 2. If the 1!'; are not all integraJ then there is at least one smaller then À-2, by Lemma 7.19, and we are done. So we may assume tha.t the 1r;'s are all integral and have t.o show tha.t there are a.t least three 1!'; not equaJ to À.
But if the loca.I graph ha..'> at least m -2 components, then, since m > 4, at least one of them only ha.s two distinct eigenvalues À and -b+ -1, hence is complete, contradiction. 0 Lemma 7 .21 Let r be a dislance-regulnr graph such that eigenvalue 8d has multiplicity m. IJk;:::: 2m and b-::::: -2 then one of the following holdS.
Proof. Let x be a vertex off. Then U1e complement of the subgra.ph f(x) has smallest eigenva.lue -2. So we can apµly Theor<'m 6.16 a.nd find the four possibilities, beca.use if ,\ = 1, then by Proposition 7.5 we find 2m 2 ~k.
If II is the line graph of a graph ~. then the maxima! size of a clique in II is at least l/2 + 1, where lis the maxima! valency in 11. lf r( x) is the complement of a line graph then the maxima! size of a coclique in r( x) is at least (k -,\ -1)/2 + L So we find m ;::: (k -,\ -1)/2 + 1 (by Proposition 7.4) and hence k -,\ + 1 $ 2m. D Before we give a. chara.cterisa.tion of the cubes we need the following lemina. We denote by C(x,y) the set {z 1 d(x, z) + d(y, z) = d(x, y)}. Lemma 7.22 Let r be a coconnected distance-regular graph with eigenvalue with multiplicity m 2: 3. Let t dim(C(x,y)), where d(x,y) = i. Then bm-t-i $ 1.
Proof. This lemma is a direct consequente of Proposition 7 .6. D Lemma 7 .23 Let r be a coconnected distance-regiilar graph with an induced quadrangle K 2 , 2 and an induced 1(2, 1 ,1 and let 8 be an eigenvalue of r of multi11licity m, m 2: 3.
Proof. Let G be the Gram matrix of a. quadrangle and H the Gram matrix of K 2 , 1 , 1 corresponding to an eigenvalue (J distinct from k. Then at least one of them has full rank. (Suppose not. Then u 2 u 1 and hence (J = -1 and it follows tha.t k = 3 and d 2: 2, contradiction.). It follows now from Lemma 7.22 that bm-2 = 1. D Theorem 7.24 Let r be a distance-regular graph with diameter d 2: 3. Jf r has an eigenvalue fJ with multiplicity d and C2 2: 2, then f is the icosa/iedmn or a hypercube.
There are now two cases, r ha.s induced qua.drangles or not. Suppose that r has an induced quadrangle. By Theorem 1.2 we have d<~. -,\ +2 By Lemma 7.23 we have that r has no induced K2.l,t· By Proposition 7.5 we have k,\ $ (,\ + l)d.
So ,\ = O. But then k = m and thus r is a. hypercube.
Suppose now that r has no induced quadra.ngles. But then by Theorem 1.16.3 of [19], the local gra.ph is a strongly regular graph with µ = 1. If the local gra.ph ha.s 3-claws, then by Lemma. 7.22 we have dim(C(.?:,y)) = 4 if d(x,y) = 2. So no loca.l graph has induced 3-claws. By Theorem 1.2.3 of [19] it follows that r is locaJly the pentagon and so f is the icosahedron. D 97 7 .2 On distance-regular graphs with ad equal to 0 The main goal of this section is to give better lower bounds on the multiplicity of a.n eigenvalue in a distance-regular graph with ad = 0.  This int<'l'section array lias eigenvaJue 11( 1 + y"2µTI) of multiplicity µ(2µ + 1 ). It is straightforward to see that 1~:~µ 21 '~~ is not an algebraic integer. Using the fact that the prodm't a.nd the snm of two algehra.k integers are aJgebrak integers, the assumption that i:;~µ 2 µ!~ is an algebra.ic integer implies that µ!l is an algehraic integer, which is only true for Jt == 1.
Remarks. (i) If distance-regnlar graphs with these arrays existed, then they would have two P-J>olynomial and two Q-polynomia.J structures.
(ii) On page 2-17 of Brouwer, Cohen and Neumaier, [19], the authors conjecture that Q-polynomial distance-regnlar graphs have at most two Q-polynomiaJ structures and all eigcnvalues are integral if th<• diameh•r is greater than or cqual to three, and not four. The reason U1at diameter 4 wa.<i <'xdud<>d wa.R the a.hove sNi<'s of intersection arrays. 98 7.3 Small multiplicities By Theorem 7.9, for given m ;;:: 3 there are only finitely many distance-regular graphs -with diameter at least 3 and with a.n eigenvalue of multiplicity m. For m = 3 those are the five platonic solids. In the first subsection we will discuss the results of ZHu [113,114] and MARTIN & ZHU [75]. In the second subsection we will discuss the algorithm we used for the classification of the distance-regular graphs with an eigenvalue of multiplicity 8. In the last subsection we shall give the classification of these graphs. If m is an even integer, then also the f ollowing graphs hm•e an eigenvalue of multiplicity m.