3-regularity in generalized quadrangles of order (s, s2)

J. A. THAS

3-REGULARITY

IN GENERALIZED OF ORDER

QUADRANGLES

(s,s 2)

ABSTRACT. If (x,y,z) is a 3-regular triad of a generalized quadrangle S=(P,B,I) of order (s, s2), s even, then {x,y,z}lu {x, y,z} ±± is contained in a subquadrangle of order s. As an application it is proved that a generalized quadrangle of order (4, 16) with at least one 3-regular triad is isomorphic to the classical generalized quadrangle Q(5,4) of order (4, 16). l. INTRODUCTION For notations and terminology we refer to [9]. Let S = (P, B, I) be a generalized quadrangle (GQ) of order (s, s2), s > 1. Then for any triad (x, y, z), i.e. for any triple (x, y, z) of pairwise noncollinear points, we have I{x, y, z} ±r = s + 1 [9]. We say that the triad (x, y, z) is 3-regular provided I{x,y, z}l±[ = s + 1. The point x is 3-regular iff each triad containing x is 3-regular. Let (x,y,z) be a 3-regular triad, let X = {x,y,z} ±, and let Y = {x,y,z} ±'. In [4] and [7] it is shown that each point of Z = P - (X u Y) is collinear with precisely two points of X u Y. The following theorems were proved by the a u t h o r ([5], [7], [8]): Let S be a G Q of order (s, s2), s > I. (a) S contains a 3-regular point iff S is isomorphic to a G Q T3(O) arising from an ovoid O of PG(3, s); (b) if s is odd, then S contains a 3-regular point iff S is isomorphic to the G Q Q(5, s) arising from the nonsingular quadric o f projective index 1 (i.e. Witt index 2) in PG(5, s); (c) if s is even, then S ~ Q(5, s) iff S contains a 3-regular point and a regular line; (d) S ~ Q(5, s) iff all point of S are 3-regular.

. T H E O R E M 1. Let (x,y,z) be a 3-regular triad of the GQ S = ( P , B , 1 ) of order (s, sZ), s > 1, and let P' be the set of all points incident with lines of the form uv, u 6 { x , y , z } ± = X and vE{x,y,z} ±± = Y. l f L is a line which is incident with no point of X u Y and if k is the number of points in P' which are incident with L, then ke{0,2} if s is odd and k e { 1 , s + 1} if s is even. Proof. Let L be a line which is incident with no point o f X w Y. If w e X = {x,y,z} ±, if w l M l m l L , and if M is not a line of the form uv, u e X and ve Y = {x, y, z} ±l, then there is just one point w ' e X - {w} which is collinear Geometriae Dedicata 17 (1984) 33-36. 0046 5755/84/0171-0033500.60 (~) 1984 by D. Reidel Publishing Company.

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with m. Hence the number r of lines uv, u e X and v~ Y, which are concurrent with L, has the parity of IX I = s + 1. Clearly r is also the n u m b e r of points in P' (P' is the set of all points incident with lines of the form uv) which are incident with L. Let {L1, L2,...} = ~ be the set of all lines which are incident with no point of X w Y, and let r~ be the n u m b e r of points in P' which are incident with L i. We have I & a l = s 3 ( s 2 - 1 ) and [ P ' - ( X u Y ) l = ( s + l ) 2 ( s - 1 ) . Clearly ~ r i = (s + 1)2(s - 1)s 2, and ~,ri(ri - 1) is the number of ordered triples (uv, u'v', Li), with u, u' distinct points of X, with v, v' distinct points of Y, and with uv ~ L~ ~ u'v' where u, v, u', v' are not incident with Li. Hence ~ri(r i - 1) -- (s + 1)2s2(s - 1). Let s be odd. Then r i is even, and so ~,ri(ri-2)>>,0 with equality iff ri~{0,2} for all i. Since Z r i ( r i - 2 ) - - ( s + 1)2sZ(s- 1 ) - ( s + 1)2(s - 1)s 2--0, we have indeed ri~{0,2 } for all i. Let s be even. T h e n ri is odd, and so ~ ( r i - 1)(ri - (s + 1)) ~< 0 with equality iff r i ~ { 1 , s + l } for all i. Since ~ , ( r i - 1 ) ( r i - ( s + l ) ) = ( s + l ) 2 s 2 ( s - 1 ) (s + 1)(s + 1)2(s - 1)s 2 + (s + 1)sa(s 2 - 1) -- 0, we have indeed ri~{1,s + 1} for all i. []

. T H E O R E M 2. Let (x, y, z) be a 3-regular triad of the GQ S = (P, B, I) of order (s, s2), s even. I f P' is ~the set of all points incident with lines of the form uv, u ~ S = {x, y,z} ± and v~ Y = {x, y,z} ±±, if B' is the set of all lines in B which are incident with at least two points of P', and if1' = I c~(P' x B'), then S' = (P', B', 1') is a subquadrangle of order s. Moreover (x, y) is a re#Mar pair of S', with {x, y}±' = {x, y, z} ± and {x, y} ±'±"= {x, y, z} ±±. Proof. We have IP'I = (s + 1)2($ - 1) + 2(s + 1) = (s + 1)(s 2 + 1). Let L be a line of B'. If L is incident with some point of X ~) Y, then clearly L is of type uv, with u e X and v~ Y. Then all points incident with L are in P'. If L is incident with no point of X u Y, then by Theorem 1, L is again incident with s + 1 points of P'. N o w by [6] S ' = (P', B', I') is a subquadrangle of order (s, t'). Since Ie'l = (s + 1)(st' + 1) we have t ' = s, and so S' is a subquadrangle of order s. Since X ~a Y c P', IX I = I Y I = s + 1, and each point of X is collinear with each point of Y, we have {x,y}±' = {x,y,z} ± and {x,y} ±'~'= {x,y,z} 1± [] .

T H E O R E M 3. l f a GQ S = (P, B, I) of order (4, 16) contains a 3-regular triad, then it is isomorphic to Q(5, 4).

3-REGULARITY IN GENERALIZED QUADRANGLES

35

Proof. Let (x, y, z) be a 3-regular triad of the G Q S of order (4, 16). Then by Theorem 2, { x , y , z } l w { x , y , z } ±± is contained in a subquadrangle S ' = (P', B', I') of order 4. By a theorem of S. E. Payne there is only one G Q of order 4 ( [3], I-5]). Hence S' may be identified with Q(4, 4), the G Q arising from a nonsingular quadric Q in PG(4, 4). In Q(4, 4) all points are regular [9]. It follows immediately that any three distinct points of a hyperbolic line of Q(4, 4) form a 3-regular triad of S. Let u be a point ofP - Q. The 17 points of Q which are collinear with u form an ovoid of Q(4,4). It is well known that each ovoid of Q(4, 4) belongs to a hyperplane PG(3, 4) of the space PG(4, 4) containing Q (this easily follows from the uniqueness of the projective plane of order 4; see also 1-1,p. 160]. There are 136 hyperplanes of PG(4, 4) which meet Q in a hyperbolic quadric, 120 which meet Q in an elliptic quadric and 85 which are tangent to Q. Thus the number of ovoids of Q(4, 4) is 120. Since for any triad (u 1 ,U2, U3) of S we have I{ul, u2, u3}ll = 5, clearly any ovoid of Q(4,4) corresponds to at most two points of P - Q. Since I P - Q r = 240, any ovoid of Q(4,4) corresponds to exactly two points of

P-Q. Consider a triad (vl, v2, v3) of S, with vieQ, i = 1,2, 3. We shall prove that (vl, v2, v3) is 3-regular. We already noticed that this is the case if v~, v2,/33 are points of a hyperbolic line of Q(4, 4). So assume that vl,/32,/33 do not belong to a common hyperbolic line. Since each point of Q(4, 4)is regular, we have {Va,v2,/33}±'= {w} in Q(4, 4) [9]. Let C be the conic Q c~ ~, where r~ is the plane Vl/32/33. Clearly w is collinear with each point of C. In Q(4, 4) there are two ovoids O, O' which contain C. The points of P - Q which correspond to O, O' are denoted by u~, u2, u'l, u~. Since Ul,U2,~/'I,U ~ are collinear with all points of C, we have {/31,V2,/33}± = {w, ul, u2, u'~, u~} and {v~, Vz, v3} ±± = C. Hence (v,, v2, v3) is 3-regular in S. Now we shall show that any point v of Q is 3-regular. If (v, v', v") is a triad consisting of points of Q, then we have already shown that (v, v', v") is 3-regular. Next, let (v, v', v") be a triad with v'eQ, v " e P - Q. Suppose that each point w~{v, v'} ±' is collinear with v". Ifw~, w2, w 3 are distinct points of {v, v'} ±', then /3"e{w~, w2, w3}L But {wa, wz, w3} ± = {v, v'} l'±', and so v"e{v, v'} ±'±' c Q, a contradiction. So let w be a point of {v,v'} J' which is not collinear with v". Further, let L be a line of Q(4, 4) through w, but distinct from vw and v'w. The points of vw, v'w, L which are collinear with v" are denoted by v~, /32,/33, respectively. By the preceding paragraph {vl, v2, v3 }±±, w, and v" are contained in a subquadrangle $1 of order 4. Clearly also v and v' are contained in S~. Interchanging the roles of Q(4, 4) and S~, we see that each triad in $1 is 3-regular. Hence (v, v',/3") is 3-regular. Finally, let (/3,v',/3") be a triad with/3', v"~P - Q. Let

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V"' be a point of Q(4,4) which is not coUinear with v or v'. Let $1 be a subquadrangle of the type described above containing v,v',v"'. Now, interchanging the roles of S 1 and Q(4,4), we know by the preceding cases that (v, v', v") is 3-regular. We conclude that v is 3-regular. Next, let u ~ P - Q. Choose a triad (u,u',u") with u',u"eQ. Then there is a subquadrangle $1 of order 4 containing u, u', u". Interchanging roles of $1 and Q(4, 4), we see that u is 3-regular. Since all points of S are 3-regular, we have S ~ Q(5, 4). [] COROLLARY. Let F be a strongly regular graph on 325 vertices with k = 68, 2 = 3 and # = 17. l f F contains a complete bipartite graph Ks, 5 as an induced subgraph, then F is the point graph of the GQ Q(5, 4). Proof. By I-2] the pseudo-geometric graph F is geometric, i.e. is the point graph of a G Q S of order (4,16). In S the triad (Xo, xl, Xz) is 3-regular, and so S ~ Q(5, 4). [] REFERENCES 1. Brouwer, A. E. and Wilbrink, H. A.: 'The Structure of Near Polygons with Quads', Geom. Dedicata 14 (1983), 145-176. 2. Cameron, P. J., Goethals J. M., and Seidel, J. J.: 'Strongly Regular Graphs Having Strongly Regular Subconstituents', d. Algebra 55 (1978), 257-280. 3. Payne, S. E.: 'Generalized Quadrangles of Order 4', I and ll, J. Comb. Theory 22 (1977), 267-279 and 280-288. 4. Payne, S. E.: 'An Inequality for Generalized Quadrangles', Proc. Amer. Math. Soc. 71 (1978), 147-152. 5.. Payne, S. E. and Thas, J. A.: "Finite generalized quadrangles', Research Notes in Mathematics, no. 110 Pitman Inc., 1984. 6. Thas, J. A.: '4-Gonal Subconfigurations of a Given 4-Gonal Configuration', Rend. Accad. Naz. Lincei 53 (1972), 520-530. 7. Thas, J. A.: 'On 4-Gonal Configurations with Parameters r = q2 + 1 and k = q + 1', I and II, Geom. Dedicata 3 (1974), 365-375; 4 (1975), 51-59. 8. Thas, J. A.: 'Combinatorial Characterizations of Generalized Quadrangles with Parameters s = q and t = q2,, Geom. Dedicata 7 (1978), 223-232. 9. Thas• J. A.: `C•mbinat•rics •f Finite Genera•ized Quadrang•es: A Survey,• Ann. Discrete Math. 14 (1982), 57-76.

Author's address: J. A. Thas, Seminar of Geometry and Combinatorics, University of Ghent, Krijgslaan 281, B-9000 Gent, Belgium (Received January 24, 1984; revised version, June 18, 1984)