On the full automorphism group of a graph


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.

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  1. [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. [2]

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

  3. [3]

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

    MATH  Article  MathSciNet  Google Scholar 

  4. [4]

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

    MATH  MathSciNet  Google Scholar 

  5. [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. [6]

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

  7. [7]

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

    MATH  MathSciNet  Google Scholar 

  8. [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. [9]

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

    Google Scholar 

  10. [10]

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

    Google Scholar 

  11. [11]

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

    MATH  Article  MathSciNet  Google Scholar 

  12. [12]

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

    Google Scholar 

  13. [13]

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

    MATH  Article  MathSciNet  Google Scholar 

  14. [14]

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

    Article  MathSciNet  Google Scholar 

  15. [15]

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

    MATH  Article  MathSciNet  Google Scholar 

  16. [16]

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

    MATH  Article  MathSciNet  Google Scholar 

  17. [17]

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

    MATH  Article  MathSciNet  Google Scholar 

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

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AMS subject classification (1980)

  • 05 C 25
  • 05 B 25