A Higman inequality for regular near polygons
Frédéric Vanhove
DOI: 10.1007/s10801-011-0275-7
Abstract
The inequality of Higman for generalized quadrangles of order ( s, t) with s>1 states that t\leq s 2. We generalize this by proving that the intersection number c i of a regular near 2 d-gon of order ( s, t) with s>1 satisfies the tight bound c i \leq ( s 2 i - 1)/( s 2 - 1), and we give properties in case of equality. It is known that hemisystems in generalized quadrangles meeting the Higman bound induce strongly regular subgraphs. We also generalize this by proving that a similar subset in regular near 2 d-gons meeting the bounds would induce a distance-regular graph with classical parameters ( d, b, α , β )=( d, - q, - ( q+1)/2, - (( - q) d +1)/2) with q an odd prime power.
Pages: 357–373
Keywords: keywords distance-regular graphs; regular near polygons; dual polar graphs; hemisystems; classical parameters
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References
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2. Bamberg, J., Giudici, M., Royle, G.: Every flock generalised quadrangle has a hemisystem. Bull. Lond. Math. Soc. 42, 795-810 (2010)
3. Bannai, E., Ito, T.: Algebraic Combinatorics. I. Association schemes. Benjamin/Cummings, Menlo Park (1984)
4. Bose, R.C., Shimamoto, T.: Classification and analysis of partially balanced incomplete block designs with two associate classes. J. Am. Stat. Assoc. 47, 151-184 (1952)
5. Bose, R.C., Shrikhande, S.S.: Geometric and pseudo-geometric graphs (q2 + 1, q + 1, 1). J. Geom. 2, 75-94 (1972)
6. Brouwer, A.E., Cohen, A.M., Neumaier, A.: Distance-Regular Graphs. Springer, Berlin (1989)
7. Brouwer, A.E., Wilbrink, H.A.: The structure of near polygons with quads. Geom. Dedic. 14, 145-176 (1983)
8. Bruen, A.A., Hirschfeld, J.W.P.: Applications of line geometry over finite fields. II. The Hermitian surface. Geom. Dedic. 7, 333-353 (1978)
9. Cameron, P.J.: Partial quadrangles. Q. J. Math. 26, 61-73 (1975)
10. Cameron, P.J., Delsarte, P., Goethals, J.M.: Hemisystems, orthogonal configurations, and dissipative conference matrices. Philips J. Res. 34, 147-162 (1979)
11. Cardinali, I., De Bruyn, B.: Regular partitions of dual polar spaces. Linear Algebra Appl. 432, 744- 769 (2010)
12. Cossidente, A., Penttila, T.: Hemisystems on the Hermitian surface. J. Lond. Math. Soc. 72, 731-741 (2005)
13. Cossidente, A., Penttila, T.: On m-regular systems on H (5, q2). J. Algebr. Comb. 29, 437-445 (2009)
14. De Bruyn, B.: Near Polygons. Birkhäuser, Basel (2006)
15. Delsarte, P.: An algebraic approach to the association schemes of coding theory. Philips Res. Rep. Suppl. 10 (1973)
16. Delsarte, P.: Association schemes and t -designs in regular semilattices. J. Comb. Theory, Ser. A 20, 230-243 (1976)
17. Delsarte, P.: Pairs of vectors in the space of an association scheme. Philips Res. Rep. 32, 373-411 (1977)
18. Feit, W., Higman, G.: The nonexistence of certain generalized polygons. J. Algebra 1, 114-131 (1964)
19. Haemers, W., Roos, C.: An inequality for generalized hexagons. Geom. Dedic. 10, 219-222 (1981)
20. Higman, D.G.: Partial geometries generalized quadrangles and strongly regular graphs. In: Atti del Convegno di Geometria Combinatoria e sua Applicazioni, Perugia, pp. 263-293 (1971)
21. Higman, D.G.: Invariant relations, coherent configurations and generalized polygons. In: Combinatorics. Math Centre Tracts, vol. 57, pp. 27-43. Math Centre, Amsterdam (1974)
22. Hiraki, A., Koolen, J.: A Higman-Haemers inequality for thick regular near polygons. J. Algebr. Comb. 20, 213-218 (2004)
23. Hiraki, A., Koolen, J.: A note on regular near polygons. Graphs Comb. 20, 485-497 (2004)
24. Hiraki, A., Koolen, J.: A generalization of an inequality of Brouwer-Wilbrink. J. Comb. Theory, Ser. A 109, 181-188 (2005)
25. Klein, A., Metsch, K., Storme, L.: Small maximal partial spreads in classical finite polar spaces. Adv. Geom. 10, 379-402 (2010)
26. Neumaier, A.: Krein conditions and near polygons. J. Comb. Theory, Ser. A 54, 201-209 (1990)
27. Pan, Y., Lu, M., Weng, C.: Triangle-free distance-regular graphs. J. Algebr. Comb. 27, 23-34 (2008)
28. Payne, S.E., Thas, J.A.: Finite Generalized Quadrangles. European Mathematical Society (EMS), Zürich (2009)
29. Segre, B.: Forme e geometrie Hermitiane, con particolare riguardo al caso finito. Ann. Mat. Pura Appl. 70, 1-201 (1965)
30. Shult, E., Yanushka, A.: Near n-gons and line systems. Geom. Dedic. 9, 1-72 (1980)
31. Stanton, D.: t -designs in classical association schemes. Graphs Comb. 2, 283-286 (1986)
32. Terwilliger, P.: Balanced sets and Q-polynomial association schemes. Graphs Comb. 4, 87-94 (1988)
33. Thas, J.A.: Ovoids and spreads of finite classical polar spaces. Geom. Dedic. 10, 135-143 (1981)
34. Thas, J.A.: Generalized quadrangles and flocks of cones. Eur. J. Comb. 8, 441-452 (1987)
35. Thas, J.A.: Interesting pointsets in generalized quadrangles and partial geometries. Linear Algebra Appl. 114/115, 103-131 (1989)
36. Thas, J.A.: Old and new results on spreads and ovoids of finite classical polar spaces. Combinatorics '90 (Gaeta, 1990). Ann. Discrete Math. 52, 529-544 (1992)
37. Tits, J.: Sur la trialité et certains groupes qui s'en déduisent. Publ. Math. 2, 13-60 (1959)
38. Vanhove, F.: Antidesigns and regularity of partial spreads in dual polar graphs. J. Comb. Des. (2010).
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