Skip to main content
SpringerLink
Log in

On the full automorphism group of a graph

  • Published:
Combinatorica Aims and scope Submit manuscript

Abstract

While it is easy to characterize the graphs on which a given transitive permutation groupG acts, it is very difficult to characterize the graphsX with Aut (X)=G. We prove here that for the certain transitive permutation groups a simple necessary condition is also sufficient. As a corollary we find that, whenG is ap-group with no homomorphism ontoZ p wrZ p , almost all Cayley graphs ofG have automorphism group isomorphic toG.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. L. Babai, On a conjecture of M. E. Watkins on graphical regular representations of groups,Compositio Math.,37 (1978), 291–296.

    MATH  MathSciNet  Google Scholar 

  2. L. Babai, Finite digraphs with given regular automorphism groups,Periodica Math. Hung. to appear.

  3. H. Bender, Transitive Gruppen gerader Ordnung, in denen jede Involution genau einen Punkt festläßt,J. Algebra,17 (1971), 527–554.

    Article  MATH  MathSciNet  Google Scholar 

  4. H. Bender, The Brauer-Suzuki-Wall theorem,Ill. J. Math.,18 (1974), 229–235.

    MATH  MathSciNet  Google Scholar 

  5. R. Brauer, M. Suzuki andG. E. Wall, A characterization of the two-dimensional unimodular projective groups over finite fields,Ill. J. Math.,2 (1958), 718–745.

    MathSciNet  Google Scholar 

  6. J. K. Doyle, Graphical Frobenius representations of abstract groups,submitted.

  7. R. Frucht, A one-regular graph of degree three,Canadian J. Math.,4 (1952), 240–247.

    MATH  MathSciNet  Google Scholar 

  8. C. D. Godsil, GRR’s for non-solvable groups,Proceedings of the conference on algebraic methods in combinatorics, Szeged (Hungary) 1978, Bolyai-North-Holland, 221–239.

  9. D. Gorenstein,Finite groups, Harper & Row, New York, (1968).

    MATH  Google Scholar 

  10. B. Huppert,Endliche Gruppen I, Springer Verlag, New York, (1967).

    MATH  Google Scholar 

  11. W. Imrich andM. E. Watkins, On automorphism groups of Cayley Graphs,Per. Math. Hung.,7 (1976), 243–258.

    Article  MATH  MathSciNet  Google Scholar 

  12. D. S. Passman,Permutation Groups, W. A. Benjamin, New York, (1968).

    MATH  Google Scholar 

  13. G. Sabidussi, Vertex-transitive graphs,Monat. Math.,68 (1964), 426–438.

    Article  MATH  MathSciNet  Google Scholar 

  14. J. Tate, Nilpotent quotient groups,Topology,3 (1964), 109–111.

    Article  MathSciNet  Google Scholar 

  15. M. E. Watkins, On the action of non-abelian groups on graphs,J. Combinatorial Theory,11 (1971), 95–104.

    Article  MATH  MathSciNet  Google Scholar 

  16. M. E. Watkins, Graphical regular representations of alternating, symmetric and miscellaneous small groups,Aequat. Math. 11 (1974), 40–50.

    Article  MATH  MathSciNet  Google Scholar 

  17. T. Yoshida, Character theoretic transfer,J. Algebra,52 (1978), 1–38.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik und Angewandte Geometrie, Mountanuniversität Leoben, Austria

    C. D. Godsil

Authors
  1. C. D. Godsil
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Godsil, C.D. On the full automorphism group of a graph. Combinatorica 1, 243–256 (1981). https://doi.org/10.1007/BF02579330

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02579330

AMS subject classification (1980)

Search

Navigation