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Ten Exceptional Geometries from Trivalent Distance Regular Graphs
Hendrik Van Maldeghem
Department of Pure Mathematics and Computer Algebra, Ghent University, Galglaan 2, 9000 Gent, Belgium
hvm@cage.rug.ac.be
Annals of Combinatorics 6 (2) p.209-228 June, 2002
AMS Subject Classification: 51E26
Abstract:
The ten distance regular graphs of valency 3and girth >4 define ten non-isomorphic neighborhood geometries, amongst which a projective plane, a generalized quadrangle, two generalized hexagons, the tilde geometry, the Desargues configuration and the Pappus configuration. All these geometries are bislim, i.e., they have three points on each line and three lines through each point. We study properties of these geometries such as embedding rank, generating rank, representation in real spaces, alternative constructions. Our main result is a general construction method for homogeneous embeddings of flag transitive self-polar bislim geometries in real projective space.
Keywords: Coxeter graph, Biggs-Smith graph, real embedding, universal embedding

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