TY - JOUR

T1 - Finite s-Arc Transitive Cayley graphs and Flag-Transitive Projective Planes

AU - Li, Cai-Heng

PY - 2005

Y1 - 2005

N2 - In this paper, a characterisation is given of finite s-arc transitive Cayley graphs with s≥2. In particular, it is shown that, for any given integer k with k≥3 and k≠7, there exists a finite set (maybe empty) of s-transitive Cayley graphs with s ε{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≥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.

AB - In this paper, a characterisation is given of finite s-arc transitive Cayley graphs with s≥2. In particular, it is shown that, for any given integer k with k≥3 and k≠7, there exists a finite set (maybe empty) of s-transitive Cayley graphs with s ε{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≥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.

U2 - 10.1090/S0002-9939-04-07549-5

DO - 10.1090/S0002-9939-04-07549-5

M3 - Article

SN - 0002-9939

VL - 133

SP - 31

EP - 41

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 1

ER -