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

Abstract

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 .

Intriguing sets of strongly regular graphs and their related structures

“…A subset I of vertices of Γ, 0 < |I| < v, is said to be intriguing with parameters (h 1 , h 2 ) if there exist constants h 1 and h 2 such that every vertex of I is adjacent to precisely h 1 vertices of I and every vertex of V (Γ) \ I is adjacent to precisely h 2 vertices of I. This concept has been introduced by Delsarte [32] in the more general framework of association schemes and investigated in different contexts by several authors [1,2,3,15,17,20,36,57]. If I is intriguing with parameters (h 1 , h 2 ), then (h 1 − h 2 − k)j I + h 2 j is an eigenvector of the adjacency matrix A with the eigenvalue h 1 − h 2 .…”

“…The graph NO − (6, 2) SRG (36,15,6,6) There are two rank three groups: PΩ − (6, 2) ≤ PGO − (6, 2) = Aut(Γ) and the eigenvalues of Γ are There are two rank three groups: PSp(4, 3) ≤ PGSp(4, 3) = Aut(Γ) and θ 1 = 2, θ 2 = −4. As for tight sets, there is one example of size 4, a line of W (3,3) and two examples of size 8: a pair of disjoint lines of W(3, 3) and the set ℓ ∪ ℓ ⊥ , where ℓ is a line of PG (3,3), that is not a line of W(3, 3).…”