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The finite vertex-primitive and vertex-biprimitive $s$-transitive graphs for $s\ge4$

Author: Cai Heng Li
Journal: Trans. Amer. Math. Soc. 353 (2001), 3511-3529
MSC (2000): Primary 05C25, 20B05
Published electronically: April 24, 2001
MathSciNet review: 1837245
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Abstract: A complete classification is given for finite vertex-primitive and vertex-biprimitive $s$-transitive graphs for $s\ge4$. The classification involves the construction of new 4-transitive graphs, namely a graph of valency 14 admitting the Monster simple group $\text{M}$, and an infinite family of graphs of valency 5 admitting projective symplectic groups $\text{PSp}(4,p)$ with $p$ prime and $p\equiv\pm1$ (mod 8). As a corollary of this classification, a conjecture of Biggs and Hoare (1983) is proved.

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Additional Information

Cai Heng Li
Affiliation: Department of Mathematics and Statistics, The University of Western Australia, Nedlands, WA 6907, Australia

Received by editor(s): November 12, 1999
Received by editor(s) in revised form: July 11, 2000
Published electronically: April 24, 2001
Additional Notes: This work forms a part of an ARC project and is supported by an ARC Fellowship
The author is grateful to C.E. Praeger, A.A. Ivanov and R. Weiss for their helpful comments on the work, and to the referee for constructive suggestions
Article copyright: © Copyright 2001 American Mathematical Society

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