Distance-regular graphs and (α, β)-geometries

We study the relation between distance-regular graphs and (α, β)-geometries in two different ways. We give necessary and sufficient conditions for the neighbourhood geometry of a distance-regular graph to be an (α, β)-geometry, and describe some (classes of) examples. On the other hand, properties of certain regular two-graphs allow us to construct (0, α)-geometries on the corresponding Taylor graphs.     

Distance-Regular (0,α)-Reguli

We introduce distance-regular (0,α)-reguli and show that they give rise to (0,α)-geometries with a distance-regular point graph. This generalises the SPG-reguli of Thas [14] and the strongly regular (α,β)-reguli of Hamilton and Mathon [9], which yield semipartial geometries and strongly regular (α,β)-geometries, respectively. We describe two infinite classes of examples, one of which is a generalisation of the well-known semipartial geometry T_n^*(B) arising from a Baer subspace PG(n, q) in PG(n, q²). Research Fellow supported by the Flemish Institute for the Promotion of Scientific and Technological Research in Industry (IWT), grant no. IWT/SB/13367/Tonesi Research assistant of the Fund for Scientific Research Flanders (FWO-Vlaanderen).