Distance-Regular (0,α)-Reguli

Abstract

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, q 2 ).

On distance-regular graphs with smallest eigenvalue at least −m

“…This partial linear space is an example of a distance-regular geometry as defined in F. De Clerck, S. De Winter, E. Kuijken and C. Tonesi [11]. The incidence graph of G is an example of a distance-semiregular graph as introduced by H. Suzuki [21].…”

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.…”