- 6 - UNIQUENESS OF A ZARA GRAPH ON 126 POINTS AND NON-EXISTENCE OF A COMPLETELY REGULAR TWO-GRAPH ON. A. Blokhuis and A.E.
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1 - 6 - UNIQUENESS OF A ZARA GRAPH ON 126 POINTS AND NON-EXISTENCE OF A COMPLETELY REGULAR TWO-GRAPH ON 288 POINTS by A. Blkhuis and A.E. Bruwer VecUc.a.:ted :t J.J. SeA-de.. n :the ec L6-tn M ftet.ulemen:t. Abstract. There is a unique graph n 126 pints satisfying the fllwing L~ree (i) (ii) cnditins: every maimal clique has si pints; fr every maimal clique C and every pint p nt in C, there are eactly tw neighburs f p in C; (iii) n pint is adjacent t all thers. Using this we shw that there eists n cmpletely regular tw-graph n 288 pints, cf. [4J, and n (287,7,3)-Zara graph, cf. [lj. 1. INTRODUCTION A Zara graph with clique sie K and neus e is a graph satisfying' (l} every maimal clique has sie K; (ii) every maimal clique has neus e (i.e., any pint nt in the clique is adjacent t eactly e pints in the clique). Fr a list f ea~ples, due t Zara, we refer t [lj and [6J~ In this nte we prve that there is nly ne Zara graph n 126 pints with clique sie 6
2 - 7 - and neus 2, which als has the prperty that n pint is adjacent t all thers. This graph, Z *, is defined as fllws: Let W be a 6-dimensinal vectr space ver GF(3), tgether with the bilinear frm <IY> = 1 Yl y. Pints f Z* are -the ne-dimensinal subspaces f W generated by a pint f nrm 1, i.e., <i> = 1. Tw such sub- 6 6 spaces are adjacent if they are rthgnal: <> ~ <y> iff <IY> = O. In the fllwing sectin Z will dente any Zara graph n 126 pints with K = 6 and e = BASIC PROPERTIES OF ZARA GRAPHS A singular subset f a Zara graph is a set f pints which is the intersectin f a cllectin f maimal cliques. Let S dente the cllectin f singular subsets. Frm [lj we qute the main therem fr Zara graphs (a graph is called aanneated if its cmpletement is cnnected) : THEOREM 1. Let G be a ccnnected Zara graph. There eists a rank functin p, S ->- IN such that (i) p (!il) = 0 (ii) If p() = i and C is a maimal clique cntaining while p E C\, then :ly E S with p (y) = i + 1 and u {p} eye C. (iii) 3r p (c) = r fr all maimal cliques C. (iv) 3R O,R 1,,Rr p () i,. is in R. maimal cliques. ~ (v) 3K O,K 1,,K r p () i.. Il = K_ ~ (vi) The graph defined n the rank sets by ~ Y i-ff i; ~ II fr all i; E X and II E Y is strngly regular.
3 - 8 - The number r is called the rank f the Zara graph. A ccnnected rank 2 Zara graph with e = 1 is essentially a generalied quadrangle. In this case singular subsets are the empty set (rank 0) the pints (rank 1) and the maimal cliques (rank 2).This graph is als dented by GQ(K-1,R -1). As an eample 1 we mentin GQ(4,2). This is a graph n 45 pints, maimal cliques have sie 5, and each pint is in three maimal cliques. This graph is unique [5J and has the fllwing descriptin: Let W be a 4-dimensinal vectr space ver GF(4) with hermitian frm <IY> = 2 1 Y Y4' where Y i = Y i Pints are the ne-dimensinal subspaces<> with <i> and <> - <y> if <IY> = 0 (and <> F <y». Anther descriptin f this graph is the fllwing: Let W be a 5-dimensinal vectr space ver GF(3) with bilinear frm <IY> = 1 Y y 5. Pints are the nedimensinal subspaces <> with <i> = 1 and <> <y> if <IY> = O. Frm the main therem n Zara graphs ne can prve: THEOREM 2. Z is a strngly regular graph, with (v,k,a,~) = (126,45,12,18). Each pint is in 27 maimal cliques, each pair f adjacent pints in 3. The induced graph n the neighburs f a given pint is (ismrphic t) GQ(4,2) A FEW RE~UUIKS ON GQ(4,2), Z * AND FISCHER SPACES The fllwing facts can be checked directly frm the descriptin f GQ(4,2) and Z* and the definitin f Z. If and yare pints at distance tw in the graph G then ~G(,y) (r just ~(,y» dentes the induced graph n the set f cmmn neighburs f and y in G.
4 - 9 - Fact 1, If f y in Z then ~(,y) is a subgraph f GQ(4,2) n 18 pints, regular with valency 3. If f y in * then ~(,y) <><3 K 3,3 Fact 2. GQ(4,2) cntains 40 subgraphs ismrphic t 3K 3,3' Thrugh each 2-claw (Le. K 1,2) in GQ(4,2) there is a unique 3 XK3,3 subgraph, even a unique K 3,3' * Let E Z. Let f(l dente the induced graph n the neighburs f, ~() the induced graph n the nn-neighburs, different frm. f() "" GQ(4,2l and each pint y E ~ () determines the subgraph Ky "" 3 K 3, 3 in r (), where Ky ~ jj(,y) Fact 3. T each subgraph K' "" 3 K 3,3' f r () there crrespnd eactly tw pints y,y' E ~(). such that Ky K y ' ; K'. Nte that y -f y'. This prperty can be used t sh\'l that * is a Fischer space.. DEFINITION. A Fischer space is a linear space (B,Ll such that (il All lines have sie 2 r 31 (iil Fr any pint, the map a : B + E, fiing and all lines thrugh, and interchanging the tw pints distinct frm n the lines f sie 3 thrugh, is an autmrphism. THEOREM 3. There is a unique Fischer space n 126 pints with 45 tw-lines n each pint 0 The prf f this fact can be fund in [2] p. 14.
5 THE UNIQUENESS PROOF, PART I Using a few lemmas, it ~lill space. By Therem 3 then Z ~ Z * be shwn that Z carries the structure f a Fischer Ntatin:. Fr a subset S f Z, we dente by S 1. the induced subgraph n the set f pints adjacent t all f S. LEMMA 1. Let {a,b,c} be a tw-claw in Z: a - b, a - c, b f c. Then {a,b,c}1. "" K3 and there is a unique pint d - a such that {a,b,c,d}.l ~ {a,b,c}.l. Mrever, d f b, d f c. Prf. Apply fact 2 t rea} "" GQ(4,2}. LEMMA 2. Let a f b in Z. Then ]l(a,b} "" 3 K 3,3 This is the main lemma; the prf will be the subject f the net sectin. 0 LEMMA 3. Let a f b in Z. There is a unique pint c E Z such that {a,b}1. {a,b,c}.l. Mrever, c f a, c f b. Prf. Cnsider a 2-claw {,y,} in ]l(a,b}. By Lemma 1 there is a pint c in {,y,}.l and c f a, c f b. By Lemma 2 ]l(a,b} "" 3K 3,3 and by fact 2 this subgraph f rea} is unique, hence ]l(a,b} = ]l(a,c} 0 THEOREM 4. Z carries the structure f a Fischer space with 126 pints and 45 tw-lines n each pint.
6 Prf. Let the tw-lines crrespnd t the edges f Z, the 3-lines t the triples {a,b,c} as in Lemma 3. This turns Z int a linear space with 45 tw-lines n each pint. It remains t be shwn that cr is an autmrphism fr all E Z. Since cr 2 = 1 it suffices t shw that y - implies cr(y) - cr(). The nly nn-trivial case is when y, E b(). Let y= r(y)nb(), Y' = f(cr(y» n b(). Then Y n Y' = ~ and jyj = jy' j 27. Since j{y,u,}ij = 6 fr all u E Y (there are three maimal cliques passing thrugh y, and has tw neighburs n each f them), and since ~(y,) = \l(cr(y),), we als have j{y,u.,rlj = 6 fr u E Yand similarly j{y,u',}ij = 6 fr u' E Y. Cunting edges between \l(,y) and b(~) it fllws that the average f j{y,u,}ij, with u E U = b(~)\(y U Y' u {y} u {cr(y)}) is 9. Cnsider an edge in ~(,y) = \l(,y'). There are three maimal cliques passing thrugh that edge, cntaining,y,y' respectively. Hence {y,u,'} is a cclique fr u E U, whence j{y,u,}ij S 9. Cmbining this yields j{y,u,}ij 9 fr all U E U. Net, cnsider a pint in Y. Since ~(,) cntradictin. Hence, Z E Y', i.e., cr(y) - cr(). This finishes the uniqueness. It remains t prve Lemma THE UNIQUENESS PROOF, PART II: PROOF OF THE MAIN LEMMA Main lerrna. Let ~ r~' in Z. Then ~ (00,00') '" 3K 3,3' The prf will be split int a number f lemmas.
7 LEMMA 4. Let S = {a,b,c,d}.be a square in Z, i.e., a - b - c - d - a and a,; c, b f d. Then Is.1l E {O,1,3}. Prf. Clearly s.1 has at mst three pints, s it suffices t shw that tw pints is impssible.1.1 Let 00,00' E S. By Lemma 1 there is a pint a' such that {d,a,b} = {a',00,00' }. Similarly there are pints b ',e',d'. If tw f the pints al,bl,e',d' cincide, then we have fund a third pint adjacent t all f S. Hence, assume they are all different. There are three maimal cliques cntaining ab One cntains c, an~ther 00', whence the third ne cntains at and b l 0 Hence a' b l,..., c~ d' - a'. Cnsidering again the clique {a,b,a',b'}, ntice that c' fa, c' - band c' - b'. It fllws that c' f a' and similarly b' f d'. The situatin is summaried in figure 1 where A = {a,a'}. Using the Zara graph prperty it fllws that the picture can be cmpleted t figure 2: ;,here E = {e,e'} etc.: Indeed, the clique {a,a',b,b'} can be cmpleted with pints figure 1 e,e'. Similarly DC can be cmpleted and {e,e'} n {f,f'} = 0. Having fund E,F,G,H, cmplete the clique {e,e',f,f'} using {i,i'}. Since i and -i' have n neighburs in A,B,C,D, they must be adjacent t G and H. Nw 00 and =' have ne neighbur in each f A,B,C,D. It fllws that bth are adjacent t i,i '. Hwever, there are three maimal figure 2
8 cliques thrugh I, tw f them already visible, whence 00 and 00' must be in the third clique. This is a cntradictin since 00 ~ 00'. The cnclusin is that a' b' c' d' is the third pint in Sl. LEMMA 5. If 00 l' 00' in Z and \1(00,00') cntains a square, then \1(00,00') "" 3K 3,3' and there is a unique pint COli such that{,l,cll}l. Prf. Let S = {a,b,c,d} be a square in \1(00,00'). Frm the previus lemma it fllws that there is a third pint, e, adjacent t the square {,a,',c}. Similarly there is a pint f adjacent t {,b,',d}, and {a,b,c,d,e,f} is a K,3 in 11(00,00'). Nw p(,') is a subgraph f r() "" GQ(4,2) with 18 pints 3 and valency 3, cntaining a K,3' This is enugh t guarantee that 11(00,00') "" 3 "" 3K,3' Let 00" be the third pcint adjacent t S. Since S is in a unique K 3 3,3 in r() it fllws that \1(00,00") = 11(00,00'). 0. I{ }ll {l LE~ 6. Let a,b,c E Z w~th a,b,c = 18. Then a,b,c} "" 3K 3,3' Prf. First nte that {a,b,c} is a cclique. Let M = {a,b,c}l and A= r(a)\m; B and C are defined similarly. Finally R = Z\ (A u B u CuM u {a} u {b} u {c}). IBI = lei = 27 24, IMI = 18. Tw adjacent pints in M have twelve cmmn neighburs, three in A,B and e and nne in R. It fllws that the neighburs f a pint r ERin M frm a cclique. A pint m E M has three neighburs in M, nine in
9 A,B and C (since Z is strngly regular with A 12). Hence m has twelve neighburs in R. Since the neighburs f r ERin M frm a cclique, r has at mst nine neighburs in M. But 9 24 = 12 18, s it is eactly nine. If M is cnnected, there are at mst tw nine ccliques in M, whence at least twelve pints f R are adjacent t the same 9-cclique. If there is an edge between tw f the twelve we have a cntradictin, if nt als Hence M is discnnected. In this case hwever, ne easily sees that M cntains a square and hence M ~ 3K 3,3 Frm nw n we will identify r() ~ GQ(4,2) with the set f istrpic pints in PG(3,4) w.r.t. a unitary frm. Fr a E ~() let Ma = ~(a,). The graph Ma has 18 vertices and is regular f valency 3. By Lemma 5, if Ma cntains a square, then Ma ~ 3K 3,3' A cmputer search fr all 18-pint subgraphs f valency 3 and girth 2: 5 f GQ(4,2) reveals that such a graph is necessarily (cnnected and) bipartite, i.e., a unin f tw vids. Nw GQ{4,2) cntains precisely tw kinds f vids, plane vids and tripd vids (cf. [3J). Let E PG(3,4)\U, where U is the set f istrpic pints. A plane vid is a set f the frm.l n u. A tripdcvid (n ) is a,set f the frm 3 U Z i n U, i=l where {'Zl'Z2'Z3} is an rthnrmal basis. On each nn-istrpic pint there are fur tripd vids. Since tw plane vids always meet, we find that each set M is ne f the fllwing (,.,here T dentes sme tripd vid n ) : a
10 (.L ~. U T ) n U {M a ~ 3K 3,3 in this case} ~ II. (.L V T ) n U where Z E.L and t T III. (T V T') n U, the unin f tw tri90d vids n the same pint..l (Nte that (T V T ) n U fr E squares, in fact K,3's.) Z and in T but nt in T des cntain If a - b, a,b E li(oo), then has tw neighburs n each f the three 6-cliques n the edge ab, s that Ma n Mb ~ 3K 2 By studying the intersectins between sets f the three types, 1,11,111 we shall see that necessarily all sets Ma are f type I. Let us prepare this study by lking at the intersectins f tw vids in GQ(4,2). A. I.L n y.l n ul 9 if y 3 if.l y j therwise. j I.L.L B. n T n ul 0 if Z r ( E and t T ).L 6 if E X and X E T Z 2 therwise. C. IT n T n ul 9 if T = T 3 if Z E 1. and ccurs in bth r nne f T X/TZ ; 0 if ( = Z and T <F T ) r ( E l. and in ne f T,Tl ; 4 if <F and I- 1. and ()l. meets TnT; therwise.
11 Net let us determine which intersectins f the sets f types I,Il,1II are f the frm 3K 2 a).l ( u T ) n (y.l U T ) n u y.l 3K iff 2 y E, i T and y t T Y b) (.L U T ) n (y.l U T ) n U "" 3K iff either ( E {y,}.l and yit) r 2 (,..L.L y U and T :3 w where w E {y,}.l). c) (.L U T ) n X (T T') n uri< 3K 2.L d) ( U T ) n (.L U T ) n U"" 3K iff either ( wand y ) ( 2 r w y w and Y E.L) r (w E.L and y = ). It fllws immediately that n set Ma can be f type III, since n type is available fr ~ when b ~ a. Each edge f ~() is in three 6-cliques and these have tw pints each in f(oo), s that we find 4-cliques in ~(). If sme 4-clique {a, b,c,d} has Ma = (.L U Ty) n U and ~ = (y.l U T ) n U (with Z E.L), then Mc and Md cannt bth be f type I (fr let {,y,,w} be an rthnrmal basis; if M c (v.l n T ) n U where v ~ v w then v E w.l and and W E Tu' v,. Tu' impssible by the definitin f a tripd); s w.l..g. M = (.L U T ) n U. c Cnsequently the three 4-cliques n the edge ab each cntain a pint c with M c (.L U T ) n U, and by Lemma 6 these three sets are distinct, s we see that the three pssibilities fr T (,. T) all ccur. Nw fiing a and c
12 and repeating the argument we find three pints b with ~ = (yi U T ) n U and similarly three pints a with Ma = (i Ty) n U and thus a subgraph ~ K 3,3.3 in ~(). But that is impssible: LEMMA 7. Let K be a subgraph f r with K ~ K 3,3,3. Then any pint utside K is adjacent t precisely three pints f K. Prf. Standard cunting arguments. Net: n 4-clique {a,b,c,d} has Ma f type II and ~, Mc' Md all f type I: Let M a (Xi U T ) n U and {,y,,w} be an rthnrmal basis; let the three y sets~, Mc and Md be (v~ U Tv) ~ are pairwise rthgnal and each n U (i = 1,2,3), then the pints v 1,v 2,v 3 is in {w,} u wi u i i 1 Of V 1 then v3 E {,y}, impssible; if v = w, v,v E wi\{} then T must cntain w v 2 and must nt cntain w, impssible; if v,v,v E (wi u i)\{w,} then we may i suppse v 2,v 3 E W \{} and the same cntradictin arises. It fllws that if a 4-clique {a,b,c,d} has Ma f type II, then there is precisely ne ther set f type II amng ~, Mc' Md - if Ma (Xi U T ) n u y i then ~ = (y u Tl n U, but a is n 27 fur-cliques and fr each f the three pssible b the edge ab is n nly 3 fur-cliques, a cntradictin. This shws that sets f type II d nt ccur at all: the main lemma is prved APPLICATIONS THEOREM 4. There des nt eist a rank 4 Zara graph G n 287 pints with clique sie 7 and neus 3.
13 Prf. Using the main L~erem fr Zara graphs it is nt difficult t shw that G is a strngly regular graph with (v,k,a,~) = (287,126,45,63). Mrever, fr each pint r E G, f(r) Z * T finish the prf we need tw lemmas. LEMMA 8. Let a ~ b in G. Then ~(a,b) is a graph n 63 pints, and fr each C E ~(a,b) we have f~(a,b) (c) "" 3K 3,3. Prf. * Cnsider fg(c) "" Z In fg(c), ~(a,b) that f~(a,b) (c) "" 3K 3,3 "" 3K 3,3' but this just means LEMMA 9. * Z des nt cntain a subgraph T n 63 pints which is lcally 3K 3,3 Prf. * Let r E Z and suppse r E T. Let K "" 3K 3,3 be the subgraph f r(r) als in T. Let figure 4 be ne f the cmpnents f K, and cnsider ft(a). We see the pints,u,v,w. a b c Since ft(a) "" 3K 3,3 there are pints 00' and cli in T als adjacent t a,u,v,w. But we knw these pints, they are unique in Z. u v w Hence:, l,c ll have precisely the same figure 4 neighburs in T. As a cnsequence the pints f T can be divided int 21 grups f 3. Let T' be the graph defined n tre 21 triples by t1 - t2 if T1 - T2 fr all T1 E t 1, T2 E t 2 Then T' is a strngly regular graph n 21 pints with k = 6, A = 1 and ~ = 1 (this is a * direct cnsequence f the structure f Z ). Nw such a graph des nt eist, since it vilates almst all knwn eistence cnditins fr strngly regular graphs. This prves the lemma and the therem.
14 THEOREM 5. There des nt eist a (nn-trivial) cmpletely regular tw-graph n 288 pints. Prf. (Fr definitins and results abut cmpletely regular tw-graphs see [4J). A cmpletely regular tw-graph n 288 pints, gives rise t at least ne rank 4 Zara graph n 287 pints with clique sie 7 and neus 3. But such a graph des nt eist by the previus therem. ACKNOWLEDGEMENTS. We are very grateful t Henny Wilbrink fr carefully reading ~he manuscript and pinting ut a "few u mistakes. REFERENCES [1J Blkhuj_s, A., Few-distance sets, thesis, T.H. Eindhven (1983). [2] Bruwer, A.E., A.M. Chen, H.A. Wilbrink, Near plygns with lines f sie three and Fischer spaces. ZW 191/83 Math. Centre, Amsterdam. [3J Bruwer, A.E., H.A. Wilbrink, Ovids and fans in the generalied quadrangle GQ(4,2), reprt ZN 102/81 Math. Centre, Amsterdam. [4J Neumaier, A., Cmpletely regular tw-graphs,arch.math. 38, (1982) [5] Seidel, J.J., Strngly regular graphs with (-1,1,0) adjacency_matri having eigenvalue 3, Linear Algebra and Applicatins (1968), [6J Zara, F., Graphes lies au espaces plaires, preprint.
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