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Strongly regular graphs
in this paper we have tried to surrmirrize the known results on strongly regular graphs. Bath pwpal and combinatorial qxcts of the theory have been included. We give the list of all knawn strongly regular graphs and a luge bibliography of this sub&t.
The theory of stron ly regular graphs (SJ. grag ose [7] in 1963, in association schemes. the rank 3 permutati combinatorial and groupal aspects oreovier, the interest in strongly reg the discovery
Contrac
X. L. Hubout
j S.rrongly regulurgruphs
here are undirected, without bcqx and mu!iiple #I and let u be the number of vertices. 9 is said to rtex is adjacent to the same number k of vertices; k is wid to be ~trmg if: o adjacent vertices x, _L the sum of the number of th s and J* and the number of vertices nonadjxvnt o nonadjacent vertices, the same sum is ccmtant. f vertices adjacent to both endpoints of an edge et of
is
vertices adjacent to two nonadjacent vertices is e set of vertkes adjacent (req. e set of vertices of c is the set of vertices is regular of valency I = V-- k - 1.
ow construct the u X u matrix A
359
In particular, if cx = 0, /3 = -.A1, af Q and it satisfies: 1
a2+33--
p+&t
y = 1, A
1s the
(0,
---I.
1)
adjacency matfix
-_ (4k-- 2’n-2~l---f)I=(v---4k-t2X+2~)J,
AJ = fl -- k)J. The eigenvalues of A are p. = 1. _k and pr
, pc,
which are roats of the
quadratic equation
The multiplicities n?t ,
m2
of
pl,
pz
are respectively,
Looking at these multiplicities, one finds necessary conditions for the existence of a stranghy regular gra fI) cl= 1T,fl=X+1 =$&Or (11) d = (A---&* + 4*(k--p) is a square and v/d divides 2k+(X--p)(v1). More;over if v is even, Ii&/d does not divide this quantity; but if v is odd, it does. In the second case, the eigenvalues p1 , p2 are odd integers. In the first CUXpQ = 0, p1 = -p2 = tfu and it is known that a necessary cundition for the existence of such raphs is that v = a2 + b*, where ILand b are integers of different parity. Graphs of this type have been constructed for ah admissible v = p”, p prime, and for some other v Let us mention that the (0, .- 1) I)-adjacency matrix graph satisfies the equation (A - p1 0 (A -- p$) = (v v- 1 +prp2 # 0, SeideI. has shown f65) that the gr Complementation with respect ts ( to a matrix A’ s
man [.34] considers the (0.1) adjacency matrix of a graph = 0, p= !, y= 0; thi!aimatrix satisfies the equation (k u_g) I= pJ a~d AJ = kL The eigenvalues s and t are ndpf bypI =--_(Zs+l),pp--(2t+l). the terminology iaf the 2-class association schemes (see Bose ces of the graph are the varieties; two varieties a~e first ciates if the two corresponding vertices are adjacent I. ?‘he correspondences between the notations are: =t+,
I=nz,
x=p;,,
2
V”‘Pll.
cdled lines such that
pair of points ties on at most one line, line contains K points, K ZB2, point is on R Itines.R > 2, a nonincident poikine pair, (p, t), there are exactly T incident with L, K > ‘1’2 I. eometry yields a strongly regl.ur graph 9 defined as
or& $$i:f the poirrt-groph of the geometry. The paramraph may be computed f~0.mthe parameters (R, K, 7’)of ~metry. We have: --I)(&-l)+T],
XL. Hubout / Stronger rcgulQr grtzphs
361
geometrizable; it is gnome trhble if such a geometry exists. A pseudo-geometric graph does not always arise from a partial geometry: for instance there exists a graph with parameters v = 16, k = 6, h = 2 which corresponds to R = 2, K = 4,T := 1 and tGs graph is not isomorphic to the point-graph of the unique partial geometry (294, 1). A theorem of Bose [?I gives a sufficient condition for a pseudogeometric graph to be geometrizable: there exists in 9 a set TJ of cliques (complete subgraphs) such that (1) two adjacent vertices are in exactly one clique of Z, (2) every vertex belongs to exactly R cliques of Z, (3) K, the common size of all Z-cliques, in greater than R. Another result of Rose gives a suf’ficient condition; a pseado-geometric graph 9 is geometrizable if
Given any partial geometry with parameters (I?, .K,T), the dual, i.e. the geometry whose points and lines are the lines ant!! the points of the Zrst one, is again a partial geometry with parameters (K, R, T). Those partial geometries have been investigated by Bose. [ 73 y Sims [ 731, Higman. [ 111 and Ahrens and Szekeres f 13. 3. tdesigns and symmetric kk$ps A t-design S,(t, k, tinis a set of u points with subsets of size k, called blocks, such that every d distinct points belong to exactly X blocks; the number b of blocks is given by IJ = X(y )/( $ ). If t = 2, it may happen that the number of blocks equals the number ofpoints. In this case X(u-- 1) = k(k- 1). The simplest examples are given by the projective spaces (the blocks are the hyperplanes). Let us mention that in a symmetric 2-design, two blocks intersect in X pants and there are k blocks containing each point. In some particular cases it is possible to derive a symmetric design from a stron ly regular graph. (A) If h = lit, take as blocks the sets
~1,the necessary conditions for the existence ?a L-1 h) = d is a square, m2 and pn divides A-;this implies T~‘IY divides X. Thus for a are finitely many strongly regular graphs with X = p# metim;“; calkd (u, k, h)-graphs, have been studied khande [ HI], Rudvalis [62] and WaHis [ $31. Rudvalis the existence af a (v, k, X) graph is equivalent to the me~LS,(2, k, v) admitting a polarity without absassary (but nat !tlfficient) condition for the existence designs) is tM (u, k,X) = (s[(s +& - 11 /a, ~&+a), m), sfs(s2-- ’ b, 3nA __ !:a is even, s and s(s2 - 1)/a must be ~~n~~~t~~n.between (u, k, A) graphs and some symmetric designs studied by Wall et al. [SO] and Hubaut 1491.
one point) in an r-dimensional euclidean space angle between every pair of lines is the same. o dete!mine the maximum number V(P)of an r-dimensional space. Seidel [ 671 proved that it is possible m a~$ symmetric (0, - 1, I)+ X u) matrix A a set of equies in an Mimensisnal space. If p is the smalkst eigenvalue muItiplieity u- .r, then the matrix (A -pI)/p ted as the grammisn matrix of u vectors in an r-space; een two lines satisfies cos ~1= I/p. in the case of strongI> raphs, the multiplicity of the shmallesteigenvalue refore the number u of equiangular lines is oreover there exists a connectisn between lar po!yhcdl*qn, so that for some sets of equithe automorphism group of the correspond-
group of G’ fixing p) E 52, has r orbits, then G’ is said to be a rank r group. In the csx where r = 3 let the 3 orbits be {p), S@J, r(p). It is obvious that q E A(/..) * p E A(@ holds if and only if (1 IS of’ every order. In this WE it is possible to deiivc from G a strongly regular graph s whose set of vertices is a; two vcrtice8 p and q are adjacew in 0 iff JJ E A(@, Nigman 234-423 has developed the theory 4 rank 3 groups; they act as an automorphism group on 9, transitive 31I the vertices and an
the edges. Infinite ck~ssesof strongly regular graphs arise ‘n ttie study of repreI sentations of classical groups, especially simple gr XI~:;..Most sf them and are a normal subi TOUTof Aut( 9). In some cases there are higher rank representations 1 ut it may happen that they yield strongly regJar graphs. A result of S&l: 1681 gives important information about the rank 3 representations of Chevaliey groups. There exists for each class of Che,valley grwp G(q) an integer N such that if y > IV, the only rank 3 representations of G(q) are representations on the cosets of ;;\parabolic subgroi:n* (In fact S&z’s result appiies to all ranks.) Such rank 3 representations do not occur for C, (y)? E7 (y), E8 (q), Fd (q) and twisted Ek (y), iF; (q) and Di (4) Most of the representations have been discovered by geometric-al mans. We would mentiorl the papers of Primrose [ SS] arid Kay-C’haudhttri [ 59) for the orthogonal groups, Bose and Chakravarti [ 8 1 and C’hakravarti [ 12$ f 31 for unitary groups, Higmarr and McLaughhn [ 43 I for sympiectic and unitarian groups and a series of papers by Wan-Zhe Sian, Yang Ben-Fu, Dai-Zong Duo and Fen-Xwning [ 19,23,85,86,87,881 have rank 3 representations
on classical groups. Exceptional representations of PO,,,,(3) have been discovered by Rudvahs [62 1. Other exceptional graphs and designs which appeal *o be related to I/,, *O;,(2) have been combinatoriahy constructed by h/mm [ 54 and &o, in cmmectiorl with coding theo;ry, by DeJs:We and Go&hals [Xl]. Taylor (761 has constructed strong graphs with PSU, (q2 1 as ~II automorphism group. For sporadic groups having rank 3 representations, we refer the reader to Tits [ 783 -
‘oulser [ 241 and
ornhoff
i I! 11
have determined the primitive rank
(!I -
esponding graphs have parameters as fb!lows: 11, (s -- 1) (g - 2) + PZ- II!), i.e., the graph is of ssible values for Qt,g) are (3’ r 4), (13,6) ) or (3L4) CT’, 10).
exist two other classes wtlen G is a subgroup of the en G acts CMa vector space 1’ such that VT2being a set of imprimitivity for GO. soup acts on an affine plane, Kailaher [ 5 1f and that the pii;lne is a translation plane. up possesses a normal regular subgroup, it is isomorphic sms of the affine space ACXn,g) containing
s determined the primitive rank 3 extensions of ural representation on k points. The following and G Z=
dihedral of order 10,
j acting on the grap e Z-transitive on both
365
Bn some cases, the knowledge sf the subdegrees k and 1 together with the rani; 3 assumption is sufficient to classify the rank 3 graphs. prow& :B] : (;Bo=d. k=c$?l - 1 J1 Fn 2 2, Sym(m) “\ Sym(2) and is of ‘type t, (rr~). (ii) ti = (T), ), m 2 5, then G is a $-fold transitive subgroup af Sym(m) and s is of type T(m), or (a) 63 = PIY.,z(8) and 9 is of type: r(s), 1 (b)p=6,m=9, 17. 27,573 (c) JI = 7, pn = 9, (d) p = &. ~YI’ = 28,36,325,903,8 128. The only E;r~~wncase is G2(.2) on 36 points CL%. (iii) u = (ZtIQn._i /Q2, k = c Q2 with Q,,, = (4’ -.- 1 )I(~~ - I), then C is a subgroup of YlY, (q) acting on the lines of Pm__1(4) transstive of the 4 simplices or possibly rra = 4 or 5 either nl = 2ar + 1 and 17 3 m 2 7 with p # (q + Pf*, An analogous resl:lt has been proved by En.cr~~io [ 221 : If u = mz and k = 3(m - I), then p = 6 t.rnlcss 1~= 4, m = fLi or 352. Furthermore if one makes the rank 3 assumption then if m >t~23, beside the two exceptional cases, G is the autom~rphlsm group of a graph of type iI&) with n = 2f. Let us mention that in these results the condition sn the parameters is independent fbf the rank 3 assumption. Another result of Higman 1351 for graphs with X = 0, JL= 1, i.e. with u = k2 + 11,is that k must be 2, 3, 7, 57. These graphs exist and are rank 3 far k = 2, 3, 7. Ashbacher 121 proved that if k = 57, there is no such rank 3 graph. IfX=Oandpf 2,4, 6, there are at most finitely many such gra
3
etersen, Clebsch alld Sch2fIi. last three f’orm a rank 3 tower. Sims ay_Chaudhuri [ SSJ cm line graphs, that -3 is df type (i)* (ii‘) or (v). nfinixly many rank 3 graphs = Zq + 1 (q = [P) and with finitely 1 and C. 11). Another foron of the smallest
) the two non-trivial orbits of k .3 representation of the same cur several times and yield a so-called rank 3 in the known tosser are generally sporadic give the paramr the reader s hat iin the first graph the sub
367
N
T
Y
cu
tin3 on the points of an hyperbolic quadric of en4 vertices = points on a generator. acting on the points of an elli ent vertices = points on $ gen acting in the points of an hermitian variety acent vertices = points on a generator. the pcrints of the X-dimensional projective repreTitan;adjacent vertices = paints WI a generator. no family of isotropic 4-planes of an hyperbolic acen t vertices ’ = 4uplanes intersecting along a he lines of an hermitian variety of PCX4,q2); intersxting lines. me representation of PSU4 (4’ ) on the lines of an hermiorphic to (4- ) with N = 3. he p’oints of a yuadric in PG(2n -- I, 2); adja0111 a rangent. on the points inside (resp. outside) a quadric of ertices = points on a nonintersecting line.
x.L.
?iubaur / Stmlg~y
regular graphs
369
tices = points on an “isotropic line” (rank 4 reprzsentation (except for d = 2, rank 3)). Same group acting on the isotropic finc!s; adjacent vertices = intersecting fines (nontransitive representation except for %= 2 rank 4). Subgroup of automorphism of AG(2, q) preserving m isotropic directions acting on the points: adjacenr vertices = points on an isotropic line. Automorphism group of a:~ hermitian parabola P in odd). If the equation of JPis x_T+ (y t j7) = 0, two vertices are adjacent iff the corresponding points of coordinates (x1 J*] $ and (.Q+_bpz j satisfy .Y1 ij 6 y 1 +y, is a square (.resp. a non-square) in GF(q) when 1 4 GFx”(q) (;esp. -*1 E CFxz(#).
1 PSL, (4) acting on an orbit of 56 complete tonics of, PG(2,4);
adjacent vertices = disjoint tonics (see Gewirtz [ 251, Goethals and Seidet [ 26) and Montague [57] ). Rank 3 representation of PSL3 (4) over Alt(6). 2. Mzs acting on the 77 blocks of S(3, 6, 22); adjacent vertices = disjoint blocks. Rank 3 representation of F&z over 24 0 3. PSU$5~) acting over subsets of autoconjugate trian with an hermitian conic. A simple construction is obtained in *he following Wiay. Let p = At, 01) be ;he set of 7 subgroups of isomorphic to A, (fixing one-letter), Let r@) be the set of 7 X b subgroups of each A6 isomorphic to A, but transitive on the 6 I Two points of r are joine if the two As intersect in D,. Se man and Singleton [48f,
x. L.
Hldxwt
= blocks interxxting
I Stmr1gly t2@JY graphs
in a single point
[ 261
l
Rank 3 rep-e-
the blocks through the mint (77) w switch graph 6 with respect to iqZI g2 )! j $X2,
~j~~~t Mock of 9c2 (see Conway
[ I?]
and also
ntation of G2 {4) on 4 16 vertices f 78].
on 31671 vertices [78]. 24 on 2048 vertices. This is related
2 *PSW, (2). in the Leech lattice
371
The lombitiatorial point of view leads alsa to mrne interesting classes of s.r. grap hs. Moreover some graphs are characterized only by combinatoria1 re!Ams. We have already mentioned the result of Seidel [[ 661 about graphs having --3 as smallest eigenvaiue. In his proof he uses the previous results sf Chang ,’ [ 13~~(3Iso a result of Connar [ 16] ), of Iioffmm [&ii1 concerning the triangular assoeiztion schemes T(rzzj and the result of Shrikh;inde [ 69 j on the L2 -association schemes L, (~‘3. In fact the parameters determine the two classes cjf graphs with 2 exceptions. ForUs 2&k= 12, X = 6, beside T(8) there exist 3 other nonisomorphic graphs [ IS 1 obtained from T(S) by the switching process [ 651. Ftlr u = 16, k = 6, A = 2 there exists a nongeometric graph with those pararneters. In the same direction Bussemaker and Seidel [ i l] have proved the existence of more than 80 nomsomoaphic graphs Lz (6), and more than 23 graphs of type .M!,, (6). Other classes of s.r. graphs have been constructed by various means. Mesner [ 561, using ;3 result of Ray-Chaudhuri [ 601, constructed two classes (in fact only one j of s.r. graphs of negative Latin square type ke., having the same parameters as L_,(n)). Bose and Shrikhande [ HI] have constructed a large number of s-r. graphs with X = p of type L(2), h’L&r) and with parameters [ 83,841 has studied graphs with (vkX)=(4r2-- I, 2r”,r2 ). I&I Wallis 1 ;A= ~1and constructed other types of graphs using affine resolvable designs; he alsa showed the existence of at least 2 nonisomorphic graphs with parameters (n2 (n + 2j, ~(n+ l), 13)for M= pQ. These graphs were first constructed by Wall 1291 when p = 2 and Ahrens and Szekeres f 11 in the general case 7% construction of and Szekeres yields another class of s.r. graphs ( .3). Ifp # 2, t struction is closely related to the construction of a graph, on an tian paraboJa in an affine plane.
particarlar there is a v .
spend to z3rank 3 representation of rtices are adjacent if the pairs con-
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II extensions when Ps2 or PSU acting in its naturaI C. 12’ , extensiorr =‘.(ii) Cy” T extension of C.2 with q = 2: (iii) (u, k, X) = ,40, 121, (2X112,30), (1X,25,8), (J26,45, !2)and
ML If 9 is a rank 3 graph with
is a graph of type C.3. The strongest antor (Rank 3 characterization of nd in a paper by A. Vanushka (a characterization
e following ones: C. ‘10’”
.
itivity of the points off a quadric in adjacent vertices = points on a non-intersectin,; line,
C. 1 19 V,,(q) vSL(5, 4) acting cm the skew-s!, rnmetric tensors; ad~acent vertices = tensars whose difference is a bivector. l
u=
1 (q2 q’O, jti= (q5 .___. )
-i
1 ) ,
l=q2(q3--1,(45.--11, x=q(q+l)(q3-2)+y---~, P.J. Camcran (communicated by $524. 2” z4 acting on M24. The vertices Of are ihe 276 unordere pairs of paints of the S(5,8,24); the vertices of I’ are the 1771 decomposititjns into 6 rlkupics. A *pair is adjacent to a set of 6 4-tuples iff it is contained in one af the 4-tuples. Given one set of 6 4tuples, there are 340 other sets in which one of the 40tuples has an intersection (3,l) c with 2 4-tuples of the other se!. 1,1=2048,
k= 276,
81 = ---41,
p, = 23
(J.H. Conway and
The author
I= 1771,
.S. Smith ).
to thank
referee
x = 44,
= 36,
the mn dxistcnce of rank thrprt permutation group af degree 3250 and smbns of rank
3 of the finite prujcctive special linear cgrOups ath. 3 (1972) 35-94.
~~l~r~~~~of ~~t~~rnorph~srns,LtJndon Math. SW. Lecture Note Series ICarn-
_Chakr~avati,
Hemitian
varieties in a finite projective space PC(N,~* ),
oto, Cksslfication and analysis o, paxtialty balancti incompktt asswidte classes, 9. Am. Statist. AQXTC. 47 (1952) 15 1-*I84. r&hande, Graphs WIwhrch each pair of vertices is adjacent to the ~ft:ther vertices, Studi W. Math. f_Aungar.S il970) 181-195. ; tr and 3 .J, Seidel, Symwtric Hadamard matrices of xder 36, Techrr.
uene~~ and nonuniqueness
of tsiangubr association schemes, Sci.
p* = 4, Sei. Record 4 61950) 12- 18.
~~~~~ 6jf order 8, W&553,61
3,086,‘?20,000,
in finite geormtrits
Bull. London Math. Sot.
ati the c0nstxuetion of in-
codes and generalixd Ziadamard matrices, fnfor-
ies in finite geometries and the ~o~~tr~ct~~~of onal geometries over finite
X. I,. IM!wur
: Strong!_vregdm graphs
379
:25) A. CewertL lihe mkpXNXS of G’(2,2, 10.56), Tr;tns.Net; yo& Acnd. f&j. 3 ] ( 1969) 656--675. 1261 JM AM& and J.J. Seidel, Quasi-symmetric block designs, proc. Ca:gary Lnt. conf., Calgary, Afta 111969)11 l- I%& [273 JM Geiathals and J-3. Seidel, Strongly regular graphs derived from combinatorial designs, Can, I. Math. 22 (?97Q) 597-614. 1281 MOHall+Aut~mor@kns of J-Jadamard matrices, SIAM J. iS.ppt. Math. 17 (6) (1969) IQ94-ilO1. I29J MefJ[& Jr., AffincGeneralized Quadrilaterals, Studies in Pure Math. (Academic Press, New York, 1971) 113-116. f 301 hi. Hail, R. hne and D, Wale-;, Designs derived from permutation groups, J. Combin. T’kmy 8 (1970) 12-22. (311 M. HaIl and D. WaJss,‘The simple group of order 604.800 J. Ugebra 9 (1968) 417-450. (32 J M.D. Hcstcsnes, Rank 3 graphs, Broc. Conf. West Mich. trrii*~, Uamazoo, Mich., 1968 (Springer, Berlin, t 969) i 9 1- 192. (331 M.D. Hegenes and D.G. Higman. Rank 3 groups and strongly reguDalrgraphs, SIAM-Arts Proe. rompltets mnAlgebra and Number Theory, Vol. I’$ I1 97 I ) 14 S- 159. (341 D.G. kligman, Finite permutation groups of rank 3, Ma&+.Z. 86 (196d ) 145 - 156. [ 35 1 D.G. Higmsn, Primitive rank 3 groups with a prime subdegree, Math, Z, 9 1 ( 1966) 70- 86. 1361 D.G. H&man, Intersection matrices for finite permutation groups, 5. Algebra 6 (1967) ’ 22-42. (371 D.G. Higman, Qn finite affine planes of rank 3, Math. 2. 104 (1968) 147- 149. (38J 0.G Higman, A survey of some questions and results about rank 3 per,nutation group, Aetes, Congres Int. Math. Rome Voi. 1 ( 1970) 361- 365 z 1391 D.G. Higman, Chtiacterization of famihcs of rank 3 permuation graphs by the subdegree I, 11, Arch. Math. 21 (1970) ;5b-156; 353- 361. 1401 D.G. Higman, Coherent coniIgur;ltions, Rend. Sem. Mat. Univ. Padova 44 (193 9,) l-26. 141f D.C. Hiaman, Partial geometries, generalized quadrangles and strongly regular graphs, Att. de\ Conk di @corn. Comb. e *+ue&@ic., Peru& 1197 1)I 263-.-294. [42f D.G. Higman, Solvability of a clas:r uf sank 3 permutation group, aGtiuya Math. .I. 4 1 (19711 89+-96, [43] D.C. Higman and J.E. MacLaughIin, Rank 3 subgraphs of finite symplectic and unitary groups, I. Reine Angew. Math. 2 18 ( 1965) 174 - 189. [44] D.G. Higman and C. Sims, A simple group of order 44,352,000, Math. 2. 105 (1968) 110-i 13. [45 ] C. Higman, On the simple group of IX. Higman ltnd CC. Sims, Illinois J* Math. 13 W6!3 74 -80. 1461 ,%.J.Hoffman, On the uniqueness o. the triartgular association scheme, 31 11960) 492--497. 1473 A.J. Hoffman and D. RayCha+ddhuri, On the line graph of a finite affine plane, @an. J. Math. 17 (1965) 687-694. hs with diarreter 2 and 3, IBM J. (481 A.J. Hoffman and R.R. Singletc n, On
rata63 in finjtc peomctrics,
Proc. Can&rid@! PlrilloS. Sot. 47 ( 195 1)
actcrizalion of line graphs, J. Combin. Theory 3 (19671 201*-2 94. he appficatwn of the geometry wf quztdratics to the cnnstructton . Carolina i 1959).
nt ta “‘The graph extent;iox theorem”.
Univ. 01. Florida, Gauwwitle,
1, 1~~~~~ graphs of I, r-type sind of’triangulti
1. I wvm281
type, Wag.
Mijth. 29
-293.
3rd Waterloo Gonf. on Combinatorics pcrmutatton
Flrz.
representations
,n isomorphic
tAc&emic
of finite ChevalIcy groups, d. Algebra 28
s~tutiofi;s of pseudo (3.5.2)
ith rank 3 ;wtc~morphism groups, J. Combin. T?wory.
and pseudo
1841 WI>. Wallis, Construction of strongly regular graphs using affinc destgns. Null. Austr. Math. SW. 4 (1971) 41 - 49. IS5 1 Warn Zhc-Xim, Studies in fmite geometries and the construc?ion of incomplete block design~s,1, fl. Some “Anzahl” theorems in symphxtic gesme:ry over finite fields. Acta. Ma~th. Sin. IS (1965) 3S4---3&f and 362 - 371. f 86 f Wan Zhe-Xian, Yang Ben-Fu, Studies in finttc gec~metrics and the construction of inwm~ plctc block designs; 111. Some “An Eahl” thetxwns in unitary geometry over finite fields and their applications. dcta Math. Sin. 15 (1965) 533-544. [87) Yang I&VI-1%.Studies in finite geomctrws and the constructian ot zncomplete blocks designs W. An zlssoci;rtion scheme %?lth many associate classes construf:ted from maxtmal completely isotropic subspa~es in sympiectic geometry over finite f*elds. Acta. Math. Sin. 15 d i965)
[SS f
812-825.
Ben-Fu. Studies in finite gwzwt,rfe s and the construction of incomplete block designs VIII. An associatio.~ scheme with many associate classes constructed from maximal conrpletcly isotropic sub\paces in unrtary geometry over finite fietdI;. Acta. Math. Sin. I5 (1965) 825-841. Yang
The theory of stron ly regular graphs (SJ. grag ose [7] in 1963, in association schemes. the rank 3 permutati combinatorial and groupal aspects oreovier, the interest in strongly reg the discovery
Contrac
X. L. Hubout
j S.rrongly regulurgruphs
here are undirected, without bcqx and mu!iiple #I and let u be the number of vertices. 9 is said to rtex is adjacent to the same number k of vertices; k is wid to be ~trmg if: o adjacent vertices x, _L the sum of the number of th s and J* and the number of vertices nonadjxvnt o nonadjacent vertices, the same sum is ccmtant. f vertices adjacent to both endpoints of an edge et of
is
vertices adjacent to two nonadjacent vertices is e set of vertkes adjacent (req. e set of vertices of c is the set of vertices is regular of valency I = V-- k - 1.
ow construct the u X u matrix A
359
In particular, if cx = 0, /3 = -.A1, af Q and it satisfies: 1
a2+33--
p+&t
y = 1, A
1s the
(0,
---I.
1)
adjacency matfix
-_ (4k-- 2’n-2~l---f)I=(v---4k-t2X+2~)J,
AJ = fl -- k)J. The eigenvalues of A are p. = 1. _k and pr
, pc,
which are roats of the
quadratic equation
The multiplicities n?t ,
m2
of
pl,
pz
are respectively,
Looking at these multiplicities, one finds necessary conditions for the existence of a stranghy regular gra fI) cl= 1T,fl=X+1 =$&Or (11) d = (A---&* + 4*(k--p) is a square and v/d divides 2k+(X--p)(v1). More;over if v is even, Ii&/d does not divide this quantity; but if v is odd, it does. In the second case, the eigenvalues p1 , p2 are odd integers. In the first CUXpQ = 0, p1 = -p2 = tfu and it is known that a necessary cundition for the existence of such raphs is that v = a2 + b*, where ILand b are integers of different parity. Graphs of this type have been constructed for ah admissible v = p”, p prime, and for some other v Let us mention that the (0, .- 1) I)-adjacency matrix graph satisfies the equation (A - p1 0 (A -- p$) = (v v- 1 +prp2 # 0, SeideI. has shown f65) that the gr Complementation with respect ts ( to a matrix A’ s
man [.34] considers the (0.1) adjacency matrix of a graph = 0, p= !, y= 0; thi!aimatrix satisfies the equation (k u_g) I= pJ a~d AJ = kL The eigenvalues s and t are ndpf bypI =--_(Zs+l),pp--(2t+l). the terminology iaf the 2-class association schemes (see Bose ces of the graph are the varieties; two varieties a~e first ciates if the two corresponding vertices are adjacent I. ?‘he correspondences between the notations are: =t+,
I=nz,
x=p;,,
2
V”‘Pll.
cdled lines such that
pair of points ties on at most one line, line contains K points, K ZB2, point is on R Itines.R > 2, a nonincident poikine pair, (p, t), there are exactly T incident with L, K > ‘1’2 I. eometry yields a strongly regl.ur graph 9 defined as
or& $$i:f the poirrt-groph of the geometry. The paramraph may be computed f~0.mthe parameters (R, K, 7’)of ~metry. We have: --I)(&-l)+T],
XL. Hubout / Stronger rcgulQr grtzphs
361
geometrizable; it is gnome trhble if such a geometry exists. A pseudo-geometric graph does not always arise from a partial geometry: for instance there exists a graph with parameters v = 16, k = 6, h = 2 which corresponds to R = 2, K = 4,T := 1 and tGs graph is not isomorphic to the point-graph of the unique partial geometry (294, 1). A theorem of Bose [?I gives a sufficient condition for a pseudogeometric graph to be geometrizable: there exists in 9 a set TJ of cliques (complete subgraphs) such that (1) two adjacent vertices are in exactly one clique of Z, (2) every vertex belongs to exactly R cliques of Z, (3) K, the common size of all Z-cliques, in greater than R. Another result of Rose gives a suf’ficient condition; a pseado-geometric graph 9 is geometrizable if
Given any partial geometry with parameters (I?, .K,T), the dual, i.e. the geometry whose points and lines are the lines ant!! the points of the Zrst one, is again a partial geometry with parameters (K, R, T). Those partial geometries have been investigated by Bose. [ 73 y Sims [ 731, Higman. [ 111 and Ahrens and Szekeres f 13. 3. tdesigns and symmetric kk$ps A t-design S,(t, k, tinis a set of u points with subsets of size k, called blocks, such that every d distinct points belong to exactly X blocks; the number b of blocks is given by IJ = X(y )/( $ ). If t = 2, it may happen that the number of blocks equals the number ofpoints. In this case X(u-- 1) = k(k- 1). The simplest examples are given by the projective spaces (the blocks are the hyperplanes). Let us mention that in a symmetric 2-design, two blocks intersect in X pants and there are k blocks containing each point. In some particular cases it is possible to derive a symmetric design from a stron ly regular graph. (A) If h = lit, take as blocks the sets
~1,the necessary conditions for the existence ?a L-1 h) = d is a square, m2 and pn divides A-;this implies T~‘IY divides X. Thus for a are finitely many strongly regular graphs with X = p# metim;“; calkd (u, k, h)-graphs, have been studied khande [ HI], Rudvalis [62] and WaHis [ $31. Rudvalis the existence af a (v, k, X) graph is equivalent to the me~LS,(2, k, v) admitting a polarity without absassary (but nat !tlfficient) condition for the existence designs) is tM (u, k,X) = (s[(s +& - 11 /a, ~&+a), m), sfs(s2-- ’ b, 3nA __ !:a is even, s and s(s2 - 1)/a must be ~~n~~~t~~n.between (u, k, A) graphs and some symmetric designs studied by Wall et al. [SO] and Hubaut 1491.
one point) in an r-dimensional euclidean space angle between every pair of lines is the same. o dete!mine the maximum number V(P)of an r-dimensional space. Seidel [ 671 proved that it is possible m a~$ symmetric (0, - 1, I)+ X u) matrix A a set of equies in an Mimensisnal space. If p is the smalkst eigenvalue muItiplieity u- .r, then the matrix (A -pI)/p ted as the grammisn matrix of u vectors in an r-space; een two lines satisfies cos ~1= I/p. in the case of strongI> raphs, the multiplicity of the shmallesteigenvalue refore the number u of equiangular lines is oreover there exists a connectisn between lar po!yhcdl*qn, so that for some sets of equithe automorphism group of the correspond-
group of G’ fixing p) E 52, has r orbits, then G’ is said to be a rank r group. In the csx where r = 3 let the 3 orbits be {p), S@J, r(p). It is obvious that q E A(/..) * p E A(@ holds if and only if (1 IS of’ every order. In this WE it is possible to deiivc from G a strongly regular graph s whose set of vertices is a; two vcrtice8 p and q are adjacew in 0 iff JJ E A(@, Nigman 234-423 has developed the theory 4 rank 3 groups; they act as an automorphism group on 9, transitive 31I the vertices and an
the edges. Infinite ck~ssesof strongly regular graphs arise ‘n ttie study of repreI sentations of classical groups, especially simple gr XI~:;..Most sf them and are a normal subi TOUTof Aut( 9). In some cases there are higher rank representations 1 ut it may happen that they yield strongly regJar graphs. A result of S&l: 1681 gives important information about the rank 3 representations of Chevaliey groups. There exists for each class of Che,valley grwp G(q) an integer N such that if y > IV, the only rank 3 representations of G(q) are representations on the cosets of ;;\parabolic subgroi:n* (In fact S&z’s result appiies to all ranks.) Such rank 3 representations do not occur for C, (y)? E7 (y), E8 (q), Fd (q) and twisted Ek (y), iF; (q) and Di (4) Most of the representations have been discovered by geometric-al mans. We would mentiorl the papers of Primrose [ SS] arid Kay-C’haudhttri [ 59) for the orthogonal groups, Bose and Chakravarti [ 8 1 and C’hakravarti [ 12$ f 31 for unitary groups, Higmarr and McLaughhn [ 43 I for sympiectic and unitarian groups and a series of papers by Wan-Zhe Sian, Yang Ben-Fu, Dai-Zong Duo and Fen-Xwning [ 19,23,85,86,87,881 have rank 3 representations
on classical groups. Exceptional representations of PO,,,,(3) have been discovered by Rudvahs [62 1. Other exceptional graphs and designs which appeal *o be related to I/,, *O;,(2) have been combinatoriahy constructed by h/mm [ 54 and &o, in cmmectiorl with coding theo;ry, by DeJs:We and Go&hals [Xl]. Taylor (761 has constructed strong graphs with PSU, (q2 1 as ~II automorphism group. For sporadic groups having rank 3 representations, we refer the reader to Tits [ 783 -
‘oulser [ 241 and
ornhoff
i I! 11
have determined the primitive rank
(!I -
esponding graphs have parameters as fb!lows: 11, (s -- 1) (g - 2) + PZ- II!), i.e., the graph is of ssible values for Qt,g) are (3’ r 4), (13,6) ) or (3L4) CT’, 10).
exist two other classes wtlen G is a subgroup of the en G acts CMa vector space 1’ such that VT2being a set of imprimitivity for GO. soup acts on an affine plane, Kailaher [ 5 1f and that the pii;lne is a translation plane. up possesses a normal regular subgroup, it is isomorphic sms of the affine space ACXn,g) containing
s determined the primitive rank 3 extensions of ural representation on k points. The following and G Z=
dihedral of order 10,
j acting on the grap e Z-transitive on both
365
Bn some cases, the knowledge sf the subdegrees k and 1 together with the rani; 3 assumption is sufficient to classify the rank 3 graphs. prow& :B] : (;Bo=d. k=c$?l - 1 J1 Fn 2 2, Sym(m) “\ Sym(2) and is of ‘type t, (rr~). (ii) ti = (T), ), m 2 5, then G is a $-fold transitive subgroup af Sym(m) and s is of type T(m), or (a) 63 = PIY.,z(8) and 9 is of type: r(s), 1 (b)p=6,m=9, 17. 27,573 (c) JI = 7, pn = 9, (d) p = &. ~YI’ = 28,36,325,903,8 128. The only E;r~~wncase is G2(.2) on 36 points CL%. (iii) u = (ZtIQn._i /Q2, k = c Q2 with Q,,, = (4’ -.- 1 )I(~~ - I), then C is a subgroup of YlY, (q) acting on the lines of Pm__1(4) transstive of the 4 simplices or possibly rra = 4 or 5 either nl = 2ar + 1 and 17 3 m 2 7 with p # (q + Pf*, An analogous resl:lt has been proved by En.cr~~io [ 221 : If u = mz and k = 3(m - I), then p = 6 t.rnlcss 1~= 4, m = fLi or 352. Furthermore if one makes the rank 3 assumption then if m >t~23, beside the two exceptional cases, G is the autom~rphlsm group of a graph of type iI&) with n = 2f. Let us mention that in these results the condition sn the parameters is independent fbf the rank 3 assumption. Another result of Higman 1351 for graphs with X = 0, JL= 1, i.e. with u = k2 + 11,is that k must be 2, 3, 7, 57. These graphs exist and are rank 3 far k = 2, 3, 7. Ashbacher 121 proved that if k = 57, there is no such rank 3 graph. IfX=Oandpf 2,4, 6, there are at most finitely many such gra
3
etersen, Clebsch alld Sch2fIi. last three f’orm a rank 3 tower. Sims ay_Chaudhuri [ SSJ cm line graphs, that -3 is df type (i)* (ii‘) or (v). nfinixly many rank 3 graphs = Zq + 1 (q = [P) and with finitely 1 and C. 11). Another foron of the smallest
) the two non-trivial orbits of k .3 representation of the same cur several times and yield a so-called rank 3 in the known tosser are generally sporadic give the paramr the reader s hat iin the first graph the sub
367
N
T
Y
cu
tin3 on the points of an hyperbolic quadric of en4 vertices = points on a generator. acting on the points of an elli ent vertices = points on $ gen acting in the points of an hermitian variety acent vertices = points on a generator. the pcrints of the X-dimensional projective repreTitan;adjacent vertices = paints WI a generator. no family of isotropic 4-planes of an hyperbolic acen t vertices ’ = 4uplanes intersecting along a he lines of an hermitian variety of PCX4,q2); intersxting lines. me representation of PSU4 (4’ ) on the lines of an hermiorphic to (4- ) with N = 3. he p’oints of a yuadric in PG(2n -- I, 2); adja0111 a rangent. on the points inside (resp. outside) a quadric of ertices = points on a nonintersecting line.
x.L.
?iubaur / Stmlg~y
regular graphs
369
tices = points on an “isotropic line” (rank 4 reprzsentation (except for d = 2, rank 3)). Same group acting on the isotropic finc!s; adjacent vertices = intersecting fines (nontransitive representation except for %= 2 rank 4). Subgroup of automorphism of AG(2, q) preserving m isotropic directions acting on the points: adjacenr vertices = points on an isotropic line. Automorphism group of a:~ hermitian parabola P in odd). If the equation of JPis x_T+ (y t j7) = 0, two vertices are adjacent iff the corresponding points of coordinates (x1 J*] $ and (.Q+_bpz j satisfy .Y1 ij 6 y 1 +y, is a square (.resp. a non-square) in GF(q) when 1 4 GFx”(q) (;esp. -*1 E CFxz(#).
1 PSL, (4) acting on an orbit of 56 complete tonics of, PG(2,4);
adjacent vertices = disjoint tonics (see Gewirtz [ 251, Goethals and Seidet [ 26) and Montague [57] ). Rank 3 representation of PSL3 (4) over Alt(6). 2. Mzs acting on the 77 blocks of S(3, 6, 22); adjacent vertices = disjoint blocks. Rank 3 representation of F&z over 24 0 3. PSU$5~) acting over subsets of autoconjugate trian with an hermitian conic. A simple construction is obtained in *he following Wiay. Let p = At, 01) be ;he set of 7 subgroups of isomorphic to A, (fixing one-letter), Let r@) be the set of 7 X b subgroups of each A6 isomorphic to A, but transitive on the 6 I Two points of r are joine if the two As intersect in D,. Se man and Singleton [48f,
x. L.
Hldxwt
= blocks interxxting
I Stmr1gly t2@JY graphs
in a single point
[ 261
l
Rank 3 rep-e-
the blocks through the mint (77) w switch graph 6 with respect to iqZI g2 )! j $X2,
~j~~~t Mock of 9c2 (see Conway
[ I?]
and also
ntation of G2 {4) on 4 16 vertices f 78].
on 31671 vertices [78]. 24 on 2048 vertices. This is related
2 *PSW, (2). in the Leech lattice
371
The lombitiatorial point of view leads alsa to mrne interesting classes of s.r. grap hs. Moreover some graphs are characterized only by combinatoria1 re!Ams. We have already mentioned the result of Seidel [[ 661 about graphs having --3 as smallest eigenvaiue. In his proof he uses the previous results sf Chang ,’ [ 13~~(3Iso a result of Connar [ 16] ), of Iioffmm [&ii1 concerning the triangular assoeiztion schemes T(rzzj and the result of Shrikh;inde [ 69 j on the L2 -association schemes L, (~‘3. In fact the parameters determine the two classes cjf graphs with 2 exceptions. ForUs 2&k= 12, X = 6, beside T(8) there exist 3 other nonisomorphic graphs [ IS 1 obtained from T(S) by the switching process [ 651. Ftlr u = 16, k = 6, A = 2 there exists a nongeometric graph with those pararneters. In the same direction Bussemaker and Seidel [ i l] have proved the existence of more than 80 nomsomoaphic graphs Lz (6), and more than 23 graphs of type .M!,, (6). Other classes of s.r. graphs have been constructed by various means. Mesner [ 561, using ;3 result of Ray-Chaudhuri [ 601, constructed two classes (in fact only one j of s.r. graphs of negative Latin square type ke., having the same parameters as L_,(n)). Bose and Shrikhande [ HI] have constructed a large number of s-r. graphs with X = p of type L(2), h’L&r) and with parameters [ 83,841 has studied graphs with (vkX)=(4r2-- I, 2r”,r2 ). I&I Wallis 1 ;A= ~1and constructed other types of graphs using affine resolvable designs; he alsa showed the existence of at least 2 nonisomorphic graphs with parameters (n2 (n + 2j, ~(n+ l), 13)for M= pQ. These graphs were first constructed by Wall 1291 when p = 2 and Ahrens and Szekeres f 11 in the general case 7% construction of and Szekeres yields another class of s.r. graphs ( .3). Ifp # 2, t struction is closely related to the construction of a graph, on an tian paraboJa in an affine plane.
particarlar there is a v .
spend to z3rank 3 representation of rtices are adjacent if the pairs con-
2 4 f
12 18 60
34 36 6 6
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II extensions when Ps2 or PSU acting in its naturaI C. 12’ , extensiorr =‘.(ii) Cy” T extension of C.2 with q = 2: (iii) (u, k, X) = ,40, 121, (2X112,30), (1X,25,8), (J26,45, !2)and
ML If 9 is a rank 3 graph with
is a graph of type C.3. The strongest antor (Rank 3 characterization of nd in a paper by A. Vanushka (a characterization
e following ones: C. ‘10’”
.
itivity of the points off a quadric in adjacent vertices = points on a non-intersectin,; line,
C. 1 19 V,,(q) vSL(5, 4) acting cm the skew-s!, rnmetric tensors; ad~acent vertices = tensars whose difference is a bivector. l
u=
1 (q2 q’O, jti= (q5 .___. )
-i
1 ) ,
l=q2(q3--1,(45.--11, x=q(q+l)(q3-2)+y---~, P.J. Camcran (communicated by $524. 2” z4 acting on M24. The vertices Of are ihe 276 unordere pairs of paints of the S(5,8,24); the vertices of I’ are the 1771 decomposititjns into 6 rlkupics. A *pair is adjacent to a set of 6 4-tuples iff it is contained in one af the 4-tuples. Given one set of 6 4tuples, there are 340 other sets in which one of the 40tuples has an intersection (3,l) c with 2 4-tuples of the other se!. 1,1=2048,
k= 276,
81 = ---41,
p, = 23
(J.H. Conway and
The author
I= 1771,
.S. Smith ).
to thank
referee
x = 44,
= 36,
the mn dxistcnce of rank thrprt permutation group af degree 3250 and smbns of rank
3 of the finite prujcctive special linear cgrOups ath. 3 (1972) 35-94.
~~l~r~~~~of ~~t~~rnorph~srns,LtJndon Math. SW. Lecture Note Series ICarn-
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