Science in China Series A: Mathematics

, Volume 47, Issue 4, pp 593–604 | Cite as

Symmetry properties of Cayley graphs of small valencies on the alternating group A5

  • Mingyao Xu
  • Shangjin Xu


The normality of symmetry property of Cayley graphs of valencies 3 and 4 on the alternating group A5 is studied. We prove that all but four such graphs are normal; that A5 is not 5-CI. A complete classification of all arc-transitive Cayley graphs on A5 of valencies 3 and 4 as well as some examples of trivalent and tetravalent GRRs of A5 is given.


Cayley graph m-CI-group normal Cayley graph arc-transitive graph GRR of a group half-transitive graph 


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  1. 1.
    Xu, M. Y., Automorphism groups and isomorphisms of Cayley digraphs, Discrete Math., 1998, 182: 309–319.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Li, C. H., Praeger, C. E., The finite simple groups with at most two fusion classes of every order, Comm. Algebra, 1996, 24: 3681–3704.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Xu, M. Y., Fang, X. G., Sim, H. S. et al., On a conjecture of Li and Praeger concerning the isomorphisms of Cayley graphs of A5, Science in China, Ser. A, 2001, 44: 1502–1508.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Li, C. H, Finite CI-groups are soluble, Bull. London Math. Soc., 1999, 31: 419–423.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Godsil, C. D., GRR’s for non-solvable groups, in Algebraic Methods in Graph Theory, Coll. Math. Soc. János Bolyai 25 (eds. Lovász, L., Sós V. T.) Amsterdam: North-Holland, 1981, 243–256.Google Scholar
  6. 6.
    Godsil, C. D., The automorphism groups of some cubic Cayley graphs, Europ. J. Combin, 1983, 4: 25–32.MATHMathSciNetGoogle Scholar
  7. 7.
    Malle, G., Saxl, J., Weigel, T., Generation of classical groups, Geom. Dedicata, 1994, 49: 85–116.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Fang, X. G., Li, C. H., Wang, J. et al., On cubic normal Cayley graphs of finite simple groups, Discrete Math., 2002, 244: 67–75.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Fang, X. G., Li, C. H., Xu, M. Y., On the automorphism group of normal edge-transitive Cayley graphs of valency 4, Preprint, 2000.Google Scholar
  10. 10.
    Li, C. H., The solution of a problem of Godsil on cubic Cayley graphs, J. Combin. Theory, Ser. B, 1998, 72: 140–142.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Li, C. H., Sim, H. S., The graphical regular representations of metacyclic p-groups, submitted.Google Scholar
  12. 12.
    Gardiner, A., Praeger, C. E., On 4-valent symmetric graphs, European J. Combin., 1994, 15: 375–381.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Gardiner, A., Praeger, C. E., A characterization of certain families of 4-valent symmetric graphs, European J. Combin., 1994, 15: 383–397.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Praeger, C. E., Xu, M. Y., A characterization of a class of symmetric graphs of twice prime valency, European J. Combin., 1989, 10: 91–102.MATHMathSciNetGoogle Scholar
  15. 15.
    Dixon, J. D., Mortimer, B., Permutation Groups, New York: Springer-Verlag, 1996.MATHGoogle Scholar
  16. 16.
    Biggs, N., Algebraic Graph Theory, 2nd. ed., Cambridge: Cambridge University Press, 1974, 1993.Google Scholar
  17. 17.
    Gorenstein, D., Finite Simple Groups, New York: Plenum Press, 1982.MATHGoogle Scholar
  18. 18.
    Conway, J. H., Curtis, R. T., Norton, S. P. et al., An Atlas of Finite Groups, Oxford: Clarendon Press, 1985.Google Scholar

Copyright information

© Science in China Press 2004

Authors and Affiliations

  • Mingyao Xu
    • 1
  • Shangjin Xu
    • 1
  1. 1.LMAM & School of Mathematical SciencesPeking UniversityBeijingChina

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