A Higman inequality for regular near polygons

The inequality of Higman for generalized quadrangles of order (s, t) with s > 1 states that t ≤ s. We will generalize this by proving that the intersection number ci of a regular near 2d-gon of order (s, t) with s > 1 satisfies the tight bound ci ≤ (s − 1)/(s − 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 will also generalize this by proving that a similar subconstituent 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) + 1)/2) with q an odd prime power.