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Finite $s$-arc transitive Cayley graphs and flag-transitive projective planes

Author: Cai Heng Li
Journal: Proc. Amer. Math. Soc. 133 (2005), 31-41
MSC (2000): Primary 20B15, 20B30, 05C25
Published electronically: July 26, 2004
MathSciNet review: 2085150
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Abstract: In this paper, a characterisation is given of finite $s$-arc transitive Cayley graphs with $s\ge2$. In particular, it is shown that, for any given integer $k$ with $k\ge3$ and $k\not=7$, there exists a finite set (maybe empty) of $s$-transitive Cayley graphs with $s\in\{3,4,5,7\}$ such that all $s$-transitive Cayley graphs of valency $k$ are their normal covers. This indicates that $s$-arc transitive Cayley graphs with $s\ge3$ are very rare. However, it is proved that there exist 4-arc transitive Cayley graphs for each admissible valency (a prime power plus one). It is then shown that the existence of a flag-transitive non-Desarguesian projective plane is equivalent to the existence of a very special arc transitive normal Cayley graph of a dihedral group.

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Additional Information

Cai Heng Li
Affiliation: School of Mathematics and Statistics, The University of Western Australia, Crawley, 6009 Western Australia, Australia

Received by editor(s): August 27, 2003
Received by editor(s) in revised form: September 11, 2003, and September 24, 2003
Published electronically: July 26, 2004
Additional Notes: This work was supported by an Australian Research Council Discovery Grant, and a QEII Fellowship. The author is grateful to the referee for his constructive comments.
Communicated by: John R. Stembridge
Article copyright: © Copyright 2004 American Mathematical Society

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