Strongly Regular (α, β)-Geometries

In this paper we introduce strongly regular (α, β)-geometries. These are a class of geometries that generalise semipartial geometries. Like semipartial geometries the underlying point graph is strongly regular and this is part of the motivation for studying the geometries. In the paper several necessary conditions for existence are given. Strongly regular (α, β)-reguli are defined, and it is shown how they may be used to construct strongly regular (α, β)-geometries. This generalises similar results by J. A. Thas in (1980, European J. Combin.1, 189–192) constructing semipartial geometries. Several constructions of strongly regular (α, β)-geometries are given, and possible parameters of existence for small cases are listed.