Journal of Algebraic Combinatorics

, Volume 23, Issue 3, pp 255–294 | Cite as

A census of semisymmetric cubic graphs on up to 768 vertices

  • Marston Conder
  • Aleksander Malnič
  • Dragan Marušič
  • Primž Potočnik


A list is given of all semisymmetric (edge- but not vertex-transitive) connected finite cubic graphs of order up to 768. This list was determined by the authors using Goldschmidt's classification of finite primitive amalgams of index (3,3), and a computer algorithm for finding all normal subgroups of up to a given index in a finitely-presented group. The list includes several previously undiscovered graphs. For each graph in the list, a significant amount of information is provided, including its girth and diameter, the order of its automorphism group, the order and structure of a minimal edge-transitive group of automorphisms, its Goldschmidt type, stabiliser partitions, and other details about its quotients and covers. A summary of all known infinite families of semisymmetric cubic graphs is also given, together with explicit rules for their construction, and members of the list are identified with these. The special case of those graphs having K 1,3 as a normal quotient is investigated in detail.


Semisymmetric graphs Edge-transitive graphs Amalgams 


  1. 1.
    W. Bosma, C. Cannon, and C. Playoust, “The Magma algebra system I: The user language,“ J. Symbolic Comput. 24 (1997), 235–265.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    I.Z. Bouwer, “An edge but not vertex transitive cubic graph,“ Bull. Can. Math. Soc. 11 (1968), 533–535.zbMATHMathSciNetGoogle Scholar
  3. 3.
    I.Z. Bouwer, “On edge but not vertex transitive regular graphs,“ J. Combin. Theory, B 12 (1972), 32–40.zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    I.Z. Bouwer (ed.), The Foster Census, Charles Babbage Research Centre, Winnipeg, 1988.zbMATHGoogle Scholar
  5. 5.
    M.D.E. Conder and P. Lorimer, “Automorphism Groups of Symmetric Graphs of Valency 3,“ J. Combin. Theory, Series B 47 (1989), 60–72.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    M.D.E. Conder, P. Dobcsányi, B. Mc Kay and G. Royle, The Extended Foster Census,
  7. 7.
    M.D.E. Conder and P. Dobcsányi, “Trivalent symmetric graphs on up to 768 vertices,“ J. Combin. Math. Combin. Comput. 40 (2002), 41–63.MathSciNetzbMATHGoogle Scholar
  8. 8.
    M.D.E. Conder, A. Malnič, D. Marušič, T. Pisanski and P. Potočnik, “The edge-transitive but not vertex-transitive cubic graph on 112 vertices”, J. Graph Theory 50 (2005), 25–42.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, and R.A. Wilson, Atlas of finite groups, Oxford University Press, Eynsham, 1985.zbMATHGoogle Scholar
  10. 10.
    J.D. Dixon and B. Mortimer, Permutation Groups, Springer-Verlag, New York, 1996.zbMATHGoogle Scholar
  11. 11.
    D.Ž. Djoković and G.L. Miller, “Regular groups of automorphisms of cubic graphs,” J. Combin. Theory. B 29 (1980), 195–230.CrossRefzbMATHGoogle Scholar
  12. 12.
    P. Dobcsányi, Home Page,
  13. 13.
    S.F. Du and D. Marušič, “Biprimitive graphs of smallest order,“ J. Algebraic Combin. 9 (1999), 151–156.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    S.F. Du and M.Y. Xu, “A classification of semisymmetric graphs of order 2pq (I),“ Comm. Algebra 28 (2000), 2685–2715. J. Combin. Theory, Series B 29 (1980), 195–230.Google Scholar
  15. 15.
    J. Folkman, “Regular line-symmetric graphs,“ J. Combin. Theory 3 (1967), 215–232.zbMATHMathSciNetGoogle Scholar
  16. 16.
    R. Frucht, “A canonical representation of trivalent Hamiltonian graphs,“ J. Graph Theory 1 (1977), 45–60.zbMATHMathSciNetGoogle Scholar
  17. 17.
    M. Giudici, C.H. Li and C.E. Praeger, “Characterising finite locally s-arc transitive graphs with a star normal quotient,“ preprint.Google Scholar
  18. 18.
    C. Godsil, “On the full automorphism group of Cayley graphs,“ Combinatorica 1 (1981), 143–156.Google Scholar
  19. 19.
    C. Godsil and G. Royle, Algebraic Graph Theory, Graduate Texts in Mathematics 207, Springer-Verlag, New York, 2001.Google Scholar
  20. 20.
    D. Goldschmidt, “Automorphisms of trivalent graphs,“ Ann. Math. 111 (1980), 377–406.zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    D. Gorenstein, Finite Groups, Harper and Row, New York, 1968.zbMATHGoogle Scholar
  22. 22.
    D. Gorenstein, Finite Simple Groups: An Introduction To Their Classification, Plenum Press, New York, 1982.zbMATHGoogle Scholar
  23. 23.
    J.L. Gross and T.W. Tucker, Topological Graph Theory, Wiley-Interscience, New York, 1987.zbMATHGoogle Scholar
  24. 24.
    M.E. Iofinova and A.A. Ivanov, Biprimitive cubic graphs, Investigations in Algebraic Theory of Combinatorial Objects (Proceedings of the seminar, Institute for System Studies, Moscow, 1985) Kluwer Academic Publishers, London, 1994, pp 459–472.Google Scholar
  25. 25.
    A.V. Ivanov, “On edge but not vertex transitive regular graphs,“ Ann. Discrete Math. 34 (1987), 273–286.zbMATHGoogle Scholar
  26. 26.
    M.H. Klin, “On edge but not vertex transitive graphs,“ Coll. Math. Soc. J. Bolyai, (25. Algebraic Methods in Graph Theory, Szeged, 1978), Budapest, 1981, 399–403.Google Scholar
  27. 27.
    F. Lazebnik and R. Viglione, “An infinite series of regular edge- but not vertex-transitive graphs,“ J. Graph Theory 41 (2002), 249–258.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    S. Lipschutz and M.Y. Xu, “Note on infinite families of trivalent semisymmetric graphs,“ European J. Combin. 23 (2002), 707–711.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    A. Malnič, D. Marušič and P. Potočnik, “On cubic graphs admitting an edge-transitive solvable group”, J. Algebraic Combinatorics 20 (2004), 99–113.CrossRefzbMATHGoogle Scholar
  30. 30.
    A. Malnič, D. Marušič, P. Potočnik and C.Q. Wang, “An infinite family of cubic edge- but not vertex-transitive graphs”, Discrete Mathematics 280 (2004), 133–148.MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    A. Malnič, D. Marušič and P. Potočnik, “Elementary abelian covers of graphs”, J. Algebraic Combinatorics 20 (2004), 71–97.CrossRefzbMATHGoogle Scholar
  32. 32.
    A. Malnič, R. Nedela, and M. Skoviera, “Lifting graph automorphisms by voltage assignments,“ European J. Combin. 21 (2000), 927–947.MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    D. Marušič, “Constructing cubic edge- but not vertex-transitive graphs,“ J. Graph Theory 35 (2000), 152–160.MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    D. Marušič and T. Pisanski, “The Gray graph revisited,“ J. Graph Theory 35 (2000), 1–7.MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    D. Marušič and P. Potočnik, “Semisymmetry of generalized Folkman graphs,“ European J. Combin. 22 (2001), 333–349.MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    M. Skoviera, “A contribution to the theory of voltage graphs,“ Discrete Math. 61 (1986), 281–292.zbMATHMathSciNetCrossRefGoogle Scholar
  37. 37.
    W.T. Tutte, “A family of cubical graphs,“ Proc. Cambridge Phil. Soc. 43 (1948), 459–474.MathSciNetCrossRefGoogle Scholar
  38. 38.
    H. Wielandt, Finite Permutation Groups, Academic Press, New York-London, 1964.zbMATHGoogle Scholar
  39. 39.
    S.E. Wilson, “A worthy family of semisymmetric graphs”, DiscreteMath. 271 (2003), 283–294.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  • Marston Conder
    • 1
  • Aleksander Malnič
    • 2
  • Dragan Marušič
    • 2
  • Primž Potočnik
    • 3
  1. 1.Department of MathematicsUniversity of AucklandAucklandNew Zealand
  2. 2.IMFM, Oddelek za matematikoUniverza v LjubljaniLjubljanaSlovenija
  3. 3.University of PrimorskaKoperSlovenia

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