A hemisystem of a nonclassical generalised quadrangle

Bamberg, John; Clerck, Frank De; Durante, Nicola

doi:10.1007/s10623-008-9251-1

Cited by 11 publications

(14 citation statements)

References 10 publications

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“…An interesting class of intriguing sets of generalized quadrangles are provided by the so-called hemisystems of generalized quadrangles of order (s 2 , s), s odd. Several new classes of such hemisystems have recently been constructed, see [1,3,8,9].…”

Section: Introductionmentioning

confidence: 99%

Intriguing Sets of Vertices of Regular Graphs

Bruyn

Suzuki

2010

Graphs and Combinatorics

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Intriguing and tight sets of vertices of point-line geometries have recently been studied in the literature. In this paper, we indicate a more general framework for dealing with these notions. Indeed, we show that some of the results obtained earlier can be generalized to larger classes of graphs. We also give some connections and relations with other notions and results from algebraic graph theory. One of the main tools in our study will be the Bose-Mesner algebra associated with the graph.

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Section: Introductionmentioning

confidence: 99%

Intriguing Sets of Vertices of Regular Graphs

Bruyn

Suzuki

2010

Graphs and Combinatorics

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“…The restriction on m was obtained for all generalized quadrangles of order (s, s 2 ) in [35]. A hemisystem in a nonclassical generalized quadrangle of order (5, 5 2 ) was constructed in [1]. Very recently, it was proved in [2] that hemisystems exist in all flock generalized quadrangles (see [34] for more information on the latter).…”

Section: Lemma 6 Ifmentioning

confidence: 99%

A Higman inequality for regular near polygons

Vanhove

2011

J Algebr Comb

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The inequality of Higman for generalized quadrangles of order (s, t) with s > 1 states that t ≤ s 2 . We generalize this by proving that the intersection number c i of a regular near 2d-gon of order (s, t) with s > 1 satisfies the tight bound c i ≤ (s 2i − 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 2d-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.We refer the reader to Sect. 2 for the definitions of, for instance, (finite) generalized polygons, near polygons, and polar spaces.Feit and Higman [18] showed that (finite) generalized n-gons of order (s, t) = (1, 1) with n ≥ 3 can only exist if n ∈ {3, 4, 6, 8, 12}; if n = 12, then s = 1 or t = 1. If s > 1, then the following inequalities must hold: if n = 4, then t ≤ s 2 [20], if n = 6, then t ≤ s 3 [19], and if n = 8, then t ≤ s 2 [21]. Bose and Shrikhande [5] also proved that if n = 4 and t = s 2 , then for any triple of nonadjacent vertices, the number

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“…Let P be a parabolic section of Q − (5, q) meeting C at two points P 1 and P 2 on the line 1 and C ⊥ at two points P 3 and P 4 on the line 2 . Then 1 , 2 meets Q − (5, q) at a 3-dimensional hyperbolic quadric and each point of n = 1 , 2 ⊥ not on planes and ⊥ yields a quadric with the required property.…”

Section: Lemma 44 the Number Of Parabolic Sections Of Q − (5 Q) Mementioning

confidence: 99%

“…Recently, Bamberg et al [1] have discovered an interesting hemisystem of the FisherThas-Walker-Kantor generalized quadrangle of order (5, 5 2 ) admitting AGL(1, 5)× S 3 , and it gives rise to…”

Section: Introduction and Basicsmentioning

confidence: 99%

Subquadrangle m-regular systems on generalized quadrangles

Cossidente

Penttila

2010

J. Combin. Designs

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Abstract:We introduce the notion of subquadrangle regular system of a generalized quadrangle. A subquadrangle regular system of order m on a generalized quadrangle of order (s, t) is a set R of embedded subquadrangles with the property that every point lies on exactly m subquadrangles of R. If m is one half of the total number of subquadrangles on a point, we call R a subquadrangle hemisystem. We construct two infinite families of symplectic subquadrangle hemisystems of the Hermitian surface H(3, q 2 ), q odd, and two infinite families of symplectic subquadrangle hemisystems of W 3 (q 2 ), q even. Some sporadic examples of symplectic subquadrangle regular systems of H(3, q 2 ) are also presented. q

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A hemisystem of a nonclassical generalised quadrangle

Cited by 11 publications

References 10 publications

Intriguing Sets of Vertices of Regular Graphs

Intriguing Sets of Vertices of Regular Graphs

A Higman inequality for regular near polygons

Subquadrangle m-regular systems on generalized quadrangles

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