A Higman inequality for regular near polygons

Abstract

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

On tetrahedrally closed line systems and a generalization of the Haemers-Roos inequality

“…The lower bound for t generalizes a lower bound obtained in [15]. In fact, the techniques we invoked in this section are similar to some of the techniques used in [15] and [32].…”

On m-ovoids of regular near polygons

“…Cameron, Goethals and Seidel [4] extended Segre's result to all generalised quadrangles of order (q, q 2 ), q odd. This was then extended further to regular near 2d-gons of order (s, t) by Vanhove [18], which also provided a generalisation of the so-called Higman bound : if s > 1 then the intersection number c i for all i ∈ {1, . .…”

“…, d} then any nontrivial m-ovoid is a hemisystem [18,Theorem 3]. Vanhove showed that for q odd if DH(2d − 1, q 2 ) has a hemisystem then it induces a distance regular graph with classical parameters [18,Theorem 4]. Hence the question of the existence of hemisystems in DH(2d −1, q 2 ) is of great interest.…”

Nonexistence of a Class of Distance-Regular Graphs

“…The properties of D-bounded distance-regular graphs were studied in [13], and these properties were used in the classification of classical distance-regular graphs of negative type [15]. Other applications of D-bounded distance-regular graphs are given in [3,12,13,15]. Before stating our main results, we show one more definition and some known results.…”