Studies of some classes of algebraically defined graphs

Abstract

The purpose of this thesis is to investigate algebraically defined graphs whose adjacency is governed by systems of polynomial equations over various fields. In certain contexts, point-line incidence graphs of finite geometries called generalized quadrangles have these algebraically defined graphs as induced subgraphs. At present, if its order is odd, there is only one known (family of) generalized quadrangle; it is unknown whether there are others. The driving question in this area is whether the known generalized quadrangles of odd order are the only ones which exist -- and thus characterize all algebraically defined graphs of this nature -- or whether there exist other non-isomorphic generalized quadrangles. Current results in the literature strongly support the belief that no new generalized quadrangles can be constructed in this way. We extend work in this field to algebraically defined graphs whose adjacency is defined by more general polynomial systems than before; our work also supports this belief.