Weighted intriguing sets of finite generalised quadrangles
John Bamberg1
, Alice Devillers1
and Jeroen Schillewaert3
1The Centre for the Mathematics of Symmetry and Computation, School of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, 6009 WA, Australia
3Department of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand
3Department of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand
DOI: 10.1007/s10801-011-0330-4
Abstract
We construct and analyse interesting integer valued functions on the points of a generalised quadrangle which lie in the orthogonal complement of a principal eigenspace of the collinearity relation. These functions generalise the intriguing sets introduced by Bamberg et al. (Combinatorica 29(1):1-17, 2009), and they provide the extra machinery to give new proofs of old results and to establish new insight into the existence of certain configurations of generalised quadrangles. In particular, we give a geometric characterisation of Payne's tight sets, we give a new proof of Thas' result that an m-ovoid of a generalised quadrangle of order ( s, s 2) is a hemisystem, and we give a bound on the values of m for which it is possible for an m-ovoid of the four dimensional Hermitian variety to exist.
Pages: 149–173
Keywords: keywords generalised quadrangle; hemisystem; $m$-ovoids; strongly regular graph
Full Text: PDF
References
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2. Bamberg, J., Giudici, M., Royle, G.F.: Every flock generalized quadrangle has a hemisystem. Bull. Lond. Math. Soc. 42(5), 795-810 (2010)
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2. Bamberg, J., Giudici, M., Royle, G.F.: Every flock generalized quadrangle has a hemisystem. Bull. Lond. Math. Soc. 42(5), 795-810 (2010)
3. Bamberg, J., Law, M., Penttila, T.: Tight sets and m-ovoids of generalised quadrangles. Combinatorica 29(1), 1-17 (2009)
4. Bose, R.C., Shrikhande, S.S.: Geometric and pseudo-geometric graphs (q2 + 1, q + 1, 1). J. Geom. 2, 75-94 (1972)
5. Brouwer, A.E., Cohen, A.M., Neumaier, A.: Distance-Regular Graphs. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol.
18. Springer, Berlin (1989)
6. Cameron, P.J.: Partial quadrangles. Q. J. Math. Oxf. Ser. 26(2), 61-73 (1975)
7. Cvetković, D.M., Doob, M., Sachs, H.: Spectra of Graphs. Pure and Applied Mathematics, vol.
87. Academic Press, New York (1980)
8. Eisfeld, J.: The eigenspaces of the Bose-Mesner algebras of the association schemes corresponding to projective spaces and polar spaces. Des. Codes Cryptogr. 17(1-3), 129-150 (1999)
9. Godsil, C.D.: Algebraic Combinatorics. Chapman and Hall Mathematics Series. Chapman & Hall, New York (1993)
10. Godsil, C., Royle, G.: Algebraic Graph Theory. Graduate Texts in Mathematics, vol.
207. Springer, New York (2001)
11. Gurobi Optimization Inc.: Gurobi optimizer version 4.0.
12. Higman, D.G.: Partial geometries, generalized quadrangles and strongly regular graphs. Atti del Convegno di Geometria Combinatoria e sue Applicazioni, Univ. Perugia, Perugia, 1970, pp. 263-293. Ist. Mat., Univ. Perugia, Perugia (1971)
13. Payne, S.E.: Tight pointsets in finite generalized quadrangles. Congr. Numer. 60, 243-260 (1987). Eighteenth Southeastern International Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, Fla., 1987)
14. Payne, S.E., Thas, J.A.: Finite Generalized Quadrangles 2nd edn. EMS Series of Lectures in Mathematics. European Mathematical Society (EMS), Zürich (2009)
15. Segre, B.: Forme e geometrie hermitiane, con particolare riguardo al caso finito. Ann. Mat. Pura Appl.
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