Intriguing sets in partial quadrangles

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

doi:10.1002/jcd.20269

Cited by 11 publications

(19 citation statements)

References 21 publications

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“…From the perspective of partial quadrangles, the following problem arose in [1]. Suppose G = (P, L) is a generalised quadrangle of order (s, s 2 ) and let p be a point of G. If we consider the set P p of points not collinear with p and the set of lines not incident with p, we produce a partial quadrangle PQ(G).…”

Section: A Generalised Quadrangle Minus a Conementioning

confidence: 99%

“…Suppose G = (P, L) is a generalised quadrangle of order (s, s 2 ) and let p be a point of G. If we consider the set P p of points not collinear with p and the set of lines not incident with p, we produce a partial quadrangle PQ(G). The point graph of PQ(G) has eigenvalues and multiplicities listed below in Table 2 (see [1]). …”

Section: A Generalised Quadrangle Minus a Conementioning

confidence: 99%

“…For the rest of the section, we examine improvements of results of Sect. 5 of [1]. For instance, we have generalised [1, Lemma 5.6] …”

Section: A Generalised Quadrangle Minus a Conementioning

confidence: 99%

“…1 Let be a line of a finite thick generalised quadrangle G of order (s, t) and let T be a group of automorphisms fixing and its concurrent lines (in the action on lines). Let m be a line concurrent but not equal to and consider the s points of m not on .…”

Section: Corollary 67 Let G Be a Generalised Quadrangle Of Order (Smentioning

confidence: 99%

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Weighted intriguing sets of finite generalised quadrangles

2011

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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.

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Section: A Generalised Quadrangle Minus a Conementioning

confidence: 99%

Section: A Generalised Quadrangle Minus a Conementioning

confidence: 99%

“…For the rest of the section, we examine improvements of results of Sect. 5 of [1]. For instance, we have generalised [1, Lemma 5.6] …”

Section: A Generalised Quadrangle Minus a Conementioning

confidence: 99%

Section: Corollary 67 Let G Be a Generalised Quadrangle Of Order (Smentioning

confidence: 99%

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Weighted intriguing sets of finite generalised quadrangles

2011

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“…Intriguing sets of vertices have been studied for several classes of strongly regular graphs which arise as collinearity graphs of point-line geometries, see [3], [11] and [12] for generalized quadrangles, [2] for polar spaces, [7] for half-spin geometries and [1] for partial quadrangles. In the present paper, we study the intriguing sets of another class of strongly regular graphs.…”

Section: · |X|mentioning

confidence: 99%

Intriguing Sets of Points of Q(2n, 2) \Q +(2n − 1, 2)

Bruyn

2011

Graphs and Combinatorics

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Intriguing sets of vertices have been studied for several classes of strongly regular graphs. In the present paper, we study intriguing sets for the graphs Γ n , n ≥ 2, which are defined as follows. Suppose Q(2n, 2), n ≥ 2, is a nonsingular parabolic quadric of PG(2n, 2) and Q + (2n − 1, 2) is a nonsingular hyperbolic quadric obtained by intersecting Q(2n, 2) with a suitable nontangent hyperplane. Then the collinearity relation of Q(2n, 2) defines a strongly regular graph Γ n on the set Q(2n, 2) \ Q + (2n − 1, 2). We describe some classes of intriguing sets of Γ n and classify all intriguing sets of Γ 2 and Γ 3 .

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