Acta Mathematicae Applicatae Sinica

, Volume 19, Issue 1, pp 83–86 | Cite as

On the Stabilizer of the Automorphism Group of a 4-valent Vertex-transitive Graph with Odd-prime-power Order

  • Yan-quan FengEmail author
  • Jin Ho Kwak
  • Ming-yao Xu
Original Papers


Let X be a 4-valent connected vertex-transitive graph with odd-prime-power order p k (k≥1), and let A be the full automorphism group of X. In this paper, we prove that the stabilizer A v of a vertex v in A is a 2-group if p ≠ 5, or a {2,3}-group if p = 5. Furthermore, if p = 5 |A v | is not divisible by 32. As a result, we show that any 4-valent connected vertex-transitive graph with odd-prime-power order p k (k≥1) is at most 1-arc-transitive for p ≠ 5 and 2-arc-transitive for p = 5.


Cayley graphs s-arc-transitive vertex-transitive 

2000 MR Subject Classification

05C25 20B25 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Baik, Y.Q., Feng, Y.Q., Sim, H.S., Xu, M.Y. On the normality of Cayley graphs of abelian groups. Algebra Colloq., 5(3): 297–304 (1998)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Biggs, N. Algebraic graph theory (Second edition). Cambridge University Press, Cambridge, 1993Google Scholar
  3. 3.
    Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilso, R.A. Atlas of finite groups. Oxford University Press, Oxford, 1985Google Scholar
  4. 4.
    Gorenstein, D. Finite simple groups. Plenum Press, New York, 1982Google Scholar
  5. 5.
    Harary, F. Graph theory. Addison-Wesley, Reading, MA, 1969Google Scholar
  6. 6.
    Huppert, B. Endliche gruppen I. Springer-Verlag, Berlin, 1967Google Scholar
  7. 7.
    Tutte, W.T. On the symmetry of cubic graphs. Canad. J. Math., 11: 621–624 (1959)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Wielandt, H. Finite permutation groups. Academic Press, New York, 1964Google Scholar
  9. 9.
    Xu, M.Y. Automorphism groups and isomorphisms of Cayley digraphs. Discrete Math., 182: 309–319 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Xu, M.Y. Half-transitive graphs of prime-power order. J. Algebraic Combinatorics, 1: 275–282 (1992)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Department of MathematicsNorthern Jiaotong UniversityBeijingChina
  2. 2.Combinatorial and Computational Mathematics CenterPohang University of Science and TechnologyPohangKorea
  3. 3.Laboratory for Mathematics and Applied Mathematics, Institute of MathematicsPeking UniversityBeijingChina

Personalised recommendations