Strongly regular (alpha,beta)-geometries

In this paper we introduce strongly regular (alpha, beta)-geometries. These are a class of geometries that generalise semipartial geometries. Like sem ipartial geometries the underlying point graph is strongly regular and this is part of the motivation for studying the geometries. In the paper severa l necessary conditions for existence are given. Strongly regular (alpha, be ta)-reguli are defined, and it is shown how they may be used to construct s trongly regular (alpha, beta)-geometries. This generalises similar results by J. A. Thas in (1980, European J. Combin. 1, 189-192) constructing semipa rtial geometries. Several constructions of strongly regular (alpha, beta)-g eometries are given, and possible parameters of existence for small cases a re listed. (C) 2001 Academic Press.