Graphs with projective subconstituents which contain short cycles
А. А. Иванов · Cambridge University Press eBooks · 1993
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 A n, D n and F 4. 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 M 22, M 23, C 02, J 4, F 2 and nonsplit extensions 3. M 22, 3 23 . C 02 and 3 4371 . F 2. 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. Introduction. The subject of the survey can be viewed in a general framework of local characterization of graphs. That is characterization by structure of the neighbourhoods of vertices. Usually this structure is described by the subgraph induced by the neighbourhood. If the vertex stabilizer in the automorphism group of the considered graph induces on the neighbourhood a doubly transitive permutation group the induced subgraph is trivial (either the complete graph or the null graph).