On limit graphs of finite vertex-primitive graphs

The class of all connected vertex-transitive graphs forms a metric space under a natural combinatorially defined metric. In this paper we study graphs which are limit points in this metric space of the subset consisting of all finite graphs that admit a vertex-primitive group of automorphisms. A description of these limit graphs provides a useful description of the possible local structures of generic finite graphs that admit a vertex-primitive automorphism group. We give an analysis of the possible types of these limit graphs, and suggest directions for future research. Some of the analysis relies on the finite simple group classification.