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CUBIC SYMMETRIC GRAPHS OF ORDER 10p

Profile image of Mohsen GhasemiMohsen Ghasemi

2013

https://doi.org/10.4134/JKMS.2013.50.2.241
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17 pages

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Abstract

An automorphism group of a graph is said to be s-regular if it acts regularly on the set of s-arcs in the graph. A graph is s-regular if its full automorphism group is s-regular. In the present paper, all sregular cubic graphs of order 10p 3 are classified for each s ≥ 1 and each prime p.

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References (41)

  1. M. Alaeiyan and M. Ghasemi, Cubic edge-transitive graphs of order 8p 2 , Bull. Austral. Math. Soc. 77 (2008), no. 2, 315-323.
  2. B. Alspach, D. Marušič, and L. Nowitz, Constructing graphs which are 1/2-transitive, J. Austral. Math. Soc. Ser. A 56 (1994), no. 3, 391-402.
  3. D. Archdeacon, P. Gvozdnjak, and J. Širan, Constructing and forbidding automorphisms in lifted maps, Math. Slovaca 47 (1997), no. 2, 113-129.
  4. D. Archdeacon, R. B. Richter, J. Širan, and M. Škoviera, Branched coverings of maps and lifts of map homomorphisms, Australas. J. Combin. 9 (1994), 109-121.
  5. Y. Cheng and J. Oxley, On weakly symmetric graphs of order twice a prime, J. Combin. Theory Ser. B 42 (1987), no. 2, 196-211.
  6. M. D. E. Conder, Trivalent (cubic) symmetric graphs on up to 2048 vertices, http: //www.math.auckland.ac.nz/ conder/ (2006).
  7. M. D. E. Conder and C. E. Praeger, Remarks on path-transitivity in finite graphs, European J. Combin. 17 (1996), no. 4, 371-378.
  8. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of Finite Groups, Clareandon Press, Oxford, 1985.
  9. D. Ž. Djoković, Automorphisms of graphs and coverings, J. Combin. Theory ser. B 16 (1974), 243-247.
  10. D. Ž. Djoković and G. L. Miller, Regular groups of automorphisms of cubic graphs, J. Combin. Theory Ser. B 29 (1980), no. 2, 195-230.
  11. S. F. Du, J. H. Kwak, and M. Y. Xu, Linear criteria for lifting automorphisms of elementary abelian regular coverings, Linear Algebra Appl. 373 (2003), 101-119.
  12. Y. Q. Feng and J. H. Kwak, Constructing an infinite family of cubic 1-regular graphs, European J. Combin. 23 (2002), no. 5, 559-565.
  13. An infinite family of cubic one-regular graphs with unsolvable automorphism groups, Discrete Math. 269 (2003), no. 1-3, 281-286.
  14. One-regular cubic graphs of order a small number times a prime or a prime square, J. Aust. Math. Soc. 76 (2004), no. 3, 345-356.
  15. Cubic symmetric graphs of order a small number times a prime or a prime square, J. Combin. Theory Ser. B 97 (2007), no. 4, 627-646.
  16. Classifying cubic symmetric graphs of order 10p or 10p 2 , Sci. China Ser. A 49 (2006), no. 3, 300-319.
  17. Y. Q. Feng, J. H. Kwak, and K. S. Wang, Classifying cubic symmetric graphs of order 8p or 8p 2 , European J. Combin. 26 (2005), no. 7, 1033-1052.
  18. Y. Q. Feng, J. H. Kwak, and M. Y. Xu, Cubic s-regular graphs of order 2p 3 , J. Graph Theory 52 (2006), no. 4, 341-352.
  19. Y. Q. Feng and K. S. Wang, s-regular cyclic coverings of the three-dimensional hypercube Q 3 , European J. Combin. 24 (2003), no. 6, 719-731.
  20. R. Frucht, A one-regular graph of degree three, Canad. J. Math. 4 (1952), 240-247.
  21. J. L. Gross and T. W. Tucker, Generating all graph coverings by permutation voltage assignment, Discrete Math. 18 (1977), no. 3, 273-283.
  22. P. Lorimer, Vertex-transitive graphs: symmetric graphs of prime valency, J. Graph The- ory 8 (1984), no. 1, 55-68.
  23. Z. P. Lu, C. Q. Wang, and M. Y. Xu, On semisymmetric cubic graphs of order 6p 2 , Sci. China Ser. A 47 (2004), no. 1, 1-17.
  24. A. Malnič, Group actions, coverings and lifts of automorphisms, Discrete Math. 182 (1998), no. 1-3, 203-218.
  25. A. Malnič and D. Marušič, Imprimitive graphs and graph coverings, in: D. Jungnickel, S. A. Vanstone (Eds.), Coding Theory, Design Theory, Group Theory: Proc. M. Hall Memorial Conf., J. Wiley and Sons, New York, 1993, pp. 221-229.
  26. A. Malnič, D. Marušič, and P. Potočnik, On cubic graphs admitting an edge-transitive solvable group, J. Algebraic Combin. 20 (2004), no. 1, 99-113.
  27. Elementary abelian covers of graphs, J. Algebraic Combin. 20 (2004), no. 1, 71-97.
  28. A. Malnič, D. Marušič, P. Potočnik, and C. Q. Wang, An infinite family of cubic edge-but not vertex-transitive graphs, Discrete Math. 280 (2004), no. 1-3, 133-148.
  29. A. Malnič, D. Marušič, and N. Seifter, Constructing infinite one-regular graphs, Euro- pean J. Combin. 20 (1999), no. 8, 845-853.
  30. A. Malnič, D. Marušič, and C. Q. Wang, Cubic edge-transitive graphs of order 2p 3 , Discrete Math. 274 (2004), no. 1-3, 187-198.
  31. A. Malnič, R. Nedela, and M. Škoviera, Lifting graph automorphisms by voltage assign- ments, European J. Combin. 21 (2000), no. 6, 927-947.
  32. A. Malnič and P. Potočnik, Invariant subspaces, duality, and covers of the Petersen graph, European J. Combin. 27 (2006), no. 6, 971-989.
  33. D. Marušič, A family of one-regular graphs of valency 4, European J. Combin. 18 (1997), no. 1, 59-64.
  34. D. Marušič and R. Nedela, Maps and half-transitive graphs of valency 4, European J. Combin. 19 (1998), no. 3, 345-354.
  35. D. Marušič and T. Pisanski, Symmetries of hexagonal molecular graphs on the torus, Croat Chemica Acta 73 (2000), 969-981.
  36. D. Marušič and M. Y. Xu, A 1/2-transitive graph of valency 4 with a nonsolvable group of automorphisms, J. Graph Theory 25 (1997), 133-138.
  37. R. C. Miller, The trivalent symmetric graphs of girth at most six, J. Combin. Theory Ser. B 10 (1971), 163-182.
  38. N. Seifter and V. I. Trofimov, Automorphism groups of covering graphs, J. Combin. Theory Ser. B 71 (1997), no. 1, 67-72.
  39. M. Škoviera, A construction to the theory of voltage groups, Discrete Math. 61 (1986), 281-292.
  40. W. T. Tutte, A family of cubical graphs, Proc. Camb. Phil. Soc. 43 (1947), 459-474.
  41. On the symmetry of cubic graphs, Canad. J. Math. 11 (1959), 621-624.

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