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

Abstract

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. (2000): 51E30, 05C12. Mathematics Subject Classification

Distance-regular graphs

“…The bipartite (point-line) incidence graph of the partial linear space is a distancesemiregular graph with girth at least 6; in fact, this gives a one-to-one correspondence between the latter type of graphs and weakly geometric distance-regular graphs. The partial linear space has also been studied by De Clerck, De Winter, Kuijken, and Tonesi [186,427] under the name distance-regular geometry.…”

Distance-Regular (0,α)-Reguli

“…One sees that it is antipodal if and only if q = 2; in this case the graph is a Taylor graph (see [3]). If the parabolic quasi-quadric is a parabolic quadric in PG(2m, 2), m ≥ 2, the geometry can also be described using the symplectic two-graph, as is done in [11].…”