Abstract

We consider graphs whose automorphism groups act transitively on the set of vertices and the stabilizer of a vertex preserves on the neighbourhood of the vertex a projective spaces structure and induces on it a flag-transitive action. Such a graph is said to possess a projective subconstituent. As an additional condition we assume that there is a short cycle in the graph (of length at most 8). The canonical examples of graphs possessing projective subconstituents and containing short cycles are related to finite Lie groups of type An,Dn and F4. There is also a remarkable “sporadic” series coming from the so-called P-geometries (geometries related to the Petersen graph). The automorphism groups of the graphs from this series are sporadic simple groups M22,M23,C02,J4,F2 and nonsplit extensions 3. M22, 323. C02 and 34371. F2. The classification of flag-transitive P-geometries was recently completed by S.V.Shpectorov and the author. The consequences of this classification for the graphs possessing projective subconstituents and containing short cycles are reported in the present survey in the context of the general situation in the subject.