There are only finitely many distance-transitive graphs with given valency > 2. This result was first shown in Cameron, Praeger, Saxl & Seitz [183] by use of the classification of finite simple groups. Below we give a proof due to Weiss [779] which is independent of this classification. A basic ingredient to the proof of Weiss’ theorem is the celebrated Thompson-Wielandt Theorem. The proof of the latter theorem requires group-theoretic preparation which can be found in Section 7.1. The Thompson-Wielandt Theorem is the content of Section 7.2 and Weiss’ theorem is in Section 7.3. Subsequently we discuss results in the cases of large girth (Section 7.4), small valency (Section 7.5), and imprimitive graphs (Section 7.6). The state of the art in overall classification and a few related results are given in the final sections (7.7–8).

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Authors and Affiliations

1. Centrum voor Wiskunde en Informatica, Stichting Mathematisch Centrum, Kruislaan 413, NL-1098 SJ, Amsterdam, The Netherlands

   Andries E. Brouwer & Arjeh M. Cohen

2. Institut für Angewandte Mathematik, Universität Freiburg, Hermann-Herder-Straße 10, D-7800, Freiburg im Breisgau, Germany

   Arnold Neumaier

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