Symmetric Designs, Sets with two Intersection Numbers and Krein Parameters of Incidence Graphs

Abstract

Let $(\mathcal{P},\mathcal{B},\mathcal{I})$ be an asymmetric $(v,k,\lambda )$ block design. The incidence graph $G$ of this design is distance-regular, hence belongs to an association scheme. In this paper, we use the algebraic structure of this association scheme to analyse certain symmetric partitions of the incidence structure.

A set with two intersection numbers is a subset $\mathcal{K}\subseteq \mathcal{P}$ with the property that $|B\cap \mathcal{K}|$ takes on only two values as ${B}$ ranges over the blocks of the design. In the special case where the design is a projective plane, these objects have received considerable attention. Two intersection theorems are proven regarding sets of this type which have a certain type of dual. Applications to the study of substructures in finite projective spaces of dimensions two and three are discussed.