On generalizing generalized polygons

Abstract

The purpose of this paper is to reveal in geometric terms a decade-old construction of certain families of graphs with nice extremal properties. Construction of the graphs in question is motivated by the way in which regular generalized polygons may be embedded in their Lie algebras, so that point-line incidence corresponds to the vanishing Lie product. The only caveat is that the generalized polygons are greatly limited in number. By performing successive truncations on an infinite root system of type ${\tilde{A}}_1$, we are able to obtain an infinite series of incidence structures which approximate the behavior of generalized polygons. Indeed, the first two members of the series are exactly the affine parts of the generalized polygons of type $A_2$ and $B_2$.