On the intersection of distance-$j$-ovoids and subpolygons in generalized polygons

Abstract

De Wispelaere and Van Maldeghem gave a technique for calculating the intersection sizes of combinatorial substructures associated with regular partitions of distance-regular graphs. This technique was based on the orthogonality of the eigenvectors which correspond to distinct eigenvalues of the (symmetric) adjacency matrix. In the present paper, we give a more general method for calculating intersection sizes of combinatorial structures. The proof of this method is based on the solution of a linear system of equations which is obtained by means of double countings. We also give a new class of regular partitions of generalized hexagons and determine under which conditions ovoids and subhexagons of order $(s',t')$ of a generalized hexagon of order $intersect in a constant number of points. If the automorphism group of the generalized hexagon is sufficiently large, then this is the case if and only if =s't'$. We derive a similar result for the intersection of distance-2-ovoids and suboctagons of generalized octagons