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Extremal generalized hexagons and triple intersection numbers
Discrete Mathematics ( IF 0.8 ) Pub Date : 2021-01-16 , DOI: 10.1016/j.disc.2020.112280
Bart De Bruyn , Frédéric Verduyn

A triple intersection number in a finite generalized hexagon S of order (s,t) is a number of the form |Γi(x)Γj(y)Γk(ω)|, where x, y are two points and ω is either a point z or a line L. The fact that S is extremal, i.e. satisfies t=s3>1, implies that certain combinatorial properties regarding triple intersection numbers hold. The earliest result in this direction is due to Haemers who showed that the number |Γi(x)Γj(y)Γk(L)| only depends on s, i, j, k and the configuration determined by (x,y,L). In this paper, we prove a similar result for the triple intersection numbers that involve three points, with exception of the case where the points are mutually opposite. In this case, we show that the 64 triple intersection numbers only depend on s and one extra parameter. We also show how Haemers’ original results can be deduced from the results obtained here, leading to a more elementary entirely combinatorial treatment where the use of any eigenvalue techniques and other non-elementary matrix theory are avoided.



中文翻译:

极值广义六边形和三重相交数

有限广义六角形中的三重相交数 小号 顺序 sŤ 是数字形式 |Γ一世XΓĴÿΓķω|,在哪里 Xÿ 有两点, ω 是一点 ž 或一条线 大号。事实小号 极端,即满足 Ť=s3>1个表示与三重交点数有关的某些组合性质成立。最早的结果是因为Haemers证明了|Γ一世XΓĴÿΓķ大号| 只取决于 s一世Ĵķ 并且配置由 Xÿ大号。在本文中,我们证明了涉及三个点的三重相交数的相似结果,但点彼此相反的情况除外。在这种情况下,我们表明64个三重交点数仅取决于s和一个额外的参数。我们还展示了如何从此处获得的结果推导出Haemers的原始结果,从而导致更基本的完全组合处理,从而避免了使用任何特征值技术和其他非元素矩阵理论。

更新日期:2021-01-18
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