Ovoids of generalized quadrangles of order and Delsarte cocliques in related strongly regular graphs

Abstract

Abstract. We investigate strongly regular graphs for which Hoffman's ratio bound and Cvetcović's inertia bound are equal. This means that ve − = m − (e − − k), where v is the number of vertices, k is the regularity, e − is the smallest eigenvalue, and m − is the multiplicity of e − . We show that Delsarte cocliques do not exist for all Taylor's 2-graphs and for point graphs of generalized quadrangles of order (q, q 2 − q) for infinitely many q. For cases where equality may hold, we show that for nearly all parameter sets, there are at most two Delsarte cocliques.

On prime order automorphisms of generalized quadrangles

“…By Lemma 3.2, Q h cannot be of type (0). By Lemma 3.3, Q h cannot be of type (1). By Lemma 3.4, Q h cannot be of type (2) or of type (2 ′ ).…”

“…Let x be an element of G of order 7. By Lemma 3.3, Q x cannot be of type (1) or of type (1 ′ ). By Lemma 3.4, Q x cannot be of type (2) or of type (2 ′ ).…”

“…We know that no automorphism of order 13 exists by Corollary 1.3, and so we assume p = 11 and let x be an element of G of order 11. By Lemma 3.2, Q x cannot be of type (0); by Lemma 3.3, Q x cannot be of type (1) or type (1 ′ ); by Lemma 3.4, Q x cannot be of type (2) or (2 ′ ); by Lemma 3.6, Q x cannot be of type (3) or (3 ′ ); and, by Lemma 3.11, Q x cannot be of type (4). Therefore, if p divides |G|, then p 7.…”

“…We are further able to use the theory that we have built up to study the automorphism groups of potential generalized quadrangles of order (4,12). The best known result thus far comes from [1], which states that if such a generalized quadrangle contains an ovoid, a set of st + 1 pairwise noncollinear points, then it cannot be point-transitive. We are able to say considerably more: Theorem 1.6.…”

“…In particular, Corollary 1.5 shows that a generalized quadrangle of order (q 2 −q, q), where q 2 −q +1 is prime, is not point-transitive. Very little is known about such (potential) generalized quadrangles; see [1]. Indeed, these results, while far from definitive, make it increasingly unlikely that such generalized quadrangles exist, since they would be very asymmetric.…”