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

Li, Cai

doi:10.1090/S0002-9939-04-07549-5

Finite $s$-arc transitive Cayley graphs and flag-transitive projective planes
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Proc. Amer. Math. Soc. 133 (2005), 31-41 Request permission

Abstract:

In this paper, a characterisation is given of finite $s$-arc transitive Cayley graphs with $s\ge 2$. In particular, it is shown that, for any given integer $k$ with $k\ge 3$ 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 $s\ge 3$ 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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