Abstract
Let Γ be a finite connected locally primitive Cayley graph of an abelian group. It is shown that one of the following holds: (1) Γ = K n , K n,n , K n,n − nK2, K n × … × K n ; (2) Γ is the standard double cover of K n × … × K n ; (3) Γ is a normal or a bi-normal Cayley graph of an elementary abelian or a meta-abelian 2-group.
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References
Alspach B, Conder M, Marušič D, et al. A classification of 2-arc-transitive circulants. J Algebraic Combin, 1996, 5: 83–86
Conway J H, Curtis R T, Norton S P, et al. Atlas of Finite Groups. London-New York: Oxford Univ Press, 1985
Dixon J D, Mortimer B. Permutation Groups. Berlin: Springer, 1991
Giudici M, Li C H, Praeger C E. Analysing finite locally s-arc transitive graphs. Trans Amer Math Soc, 2004, 356: 291–317
Godsil C D. On the full automorphism group of a graph. Combinatorial, 1981, 1: 243–256
Gorenstein D. Finite Simple Groups. New York: Plenum Press, 1982
Ivanov A A, Praeger C E. On finite affine 2-arc transitive graphs. Europ J Combin, 1993, 14: 421–444
Li C H, Praeger C E, Venkatesh A, et al. Finite locally-primitive graphs. Discrete Math, 2002, 246: 197–218
Li C H. The finite primitive permutation groups containing an abelian regular subgroup. Proc London Math Soc, 2003, 87: 725–748
Li C H. Finite edge-transitive Cayley graphs and rotary Cayley maps. Trans Amer Math Soc, 2006, 358: 4605–4635
Li C H, Pan J. Finite 2-arc transitive abelian Cayley graphs. Europ J Combin, 2008, 29: 148–158
Li C H, Pan J, Ma L. Locally primitive graphs of prime-power order. J Aust Math Soc, 2009, 86: 111–122
Marušič D, Potočnik P, Waller A O. Classifying 2-arc-transitive Cayley graphs of abelian groups with at most three involutions. Unpublished manuscript, 1998
Potočnik P. On 2-arc-transitive Cayley graphs of abelian groups. Discrete Math, 2002, 244: 417–421
Praeger C E. Primitive permutation groups with a doubly transitive subconstituent. J Aust Math Soc Ser A, 1988, 45: 66–77
Praeger C E. An O’Nan-Scott theorem for finite quasiprimitive permutation groups and an application to 2-arc transitive graphs. J London Math Soc, 1992, 47: 227–239
Karpilovsky G. The Schur Multiplier. Oxford Science Publication. Oxford: Clarendon Press, 1987
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Li, C., Lou, B. & Pan, J. Finite locally primitive abelian Cayley graphs. Sci. China Math. 54, 845–854 (2011). https://doi.org/10.1007/s11425-010-4134-0
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DOI: https://doi.org/10.1007/s11425-010-4134-0