Skip to main content
SpringerLink
Log in

Valency seven symmetric graphs of order 2pq

  • Published:
Czechoslovak Mathematical Journal Aims and scope Submit manuscript

Abstract

A graph is said to be symmetric if its automorphism group acts transitively on its arcs. In this paper, all connected valency seven symmetric graphs of order 2pq are classified, where p, q are distinct primes. It follows from the classification that there is a unique connected valency seven symmetric graph of order 4p, and that for odd primes p and q, there is an infinite family of connected valency seven one-regular graphs of order 2pq with solvable automorphism groups, and there are four sporadic ones with nonsolvable automorphism groups, which is 1, 2, 3-arc transitive, respectively. In particular, one of the four sporadic ones is primitive, and the other two of the four sporadic ones are bi-primitive.

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. W. Bosma, J. Cannon, C. Playoust: The Magma algebra system. I. The user language. J. Symb. Comput. 24 (1997), 235–265.

    Article  MathSciNet  MATH  Google Scholar 

  2. Y. Cheng, J. Oxley: On weakly symmetric graphs of order twice a prime. J. Comb. Theory, Ser. B 42 (1987), 196–211.

    Article  MathSciNet  MATH  Google Scholar 

  3. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson: Atlas of Finite Groups. Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford University Press, Eynsham, 1985.

    MATH  Google Scholar 

  4. D. Ž. Djoković, G. L. Miller: Regular groups of automorphisms of cubic graphs. J. Comb. Theory, Ser. B 29 (1980), 195–230.

    Article  MathSciNet  Google Scholar 

  5. X. G. Fang, C. E. Praeger: Finite two-arc transitive graphs admitting a Suzuki simple group. Commun. Algebra 27 (1999), 3727–3754.

    Article  MathSciNet  MATH  Google Scholar 

  6. X. Fang, J. Wang, M. Y. Xu: On 1-arc-regular graphs. Eur. J. Comb. 23 (2002), 785–791.

    Article  MathSciNet  MATH  Google Scholar 

  7. Y.-Q. Feng, M. Ghasemi, D.-W. Yang: Cubic symmetric graphs of order 8p3. Discrete Math. 318 (2014), 62–70.

    Article  MathSciNet  MATH  Google Scholar 

  8. Y.-Q. Feng, J. H. Kwak: Cubic symmetric graphs of order a small number times a prime or a prime square. J. Comb. Theory, Ser. B 97 (2007), 627–646.

    Article  MathSciNet  MATH  Google Scholar 

  9. Y.-Q. Feng, J. H. Kwak, M.-Y. Xu: Cubic s-regular graphs of order 2p3. J. Graph Theory 52 (2006), 341–352.

    Article  MathSciNet  MATH  Google Scholar 

  10. Y.-Q. Feng, Y.-T. Li: One-regular graphs of square-free order of prime valency. Eur. J. Comb. 32 (2011), 265–275.

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Gardiner, C. E. Praeger: On 4-valent symmetric graphs. Eur. J. Comb. 15 (1994), 375–381.

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Gardiner, C. E. Praeger: A characterization of certain families of 4-valent symmetric graphs. Eur. J. Comb. 15 (1994), 383–397.

    Article  MathSciNet  MATH  Google Scholar 

  13. D. Gorenstein: Finite Simple Groups. An Introduction to Their Classification. The University Series in Mathematics, Plenum Press, New York, 1982.

    MATH  Google Scholar 

  14. S.-T. Guo, Y.-Q. Feng: A note on pentavalent s-transitive graphs. Discrete Math. 312 (2012), 2214–2216.

    Article  MathSciNet  MATH  Google Scholar 

  15. S. Guo, Y. Li, X. Hua: (G, s)-transitive graphs of valency 7. Algebra Colloq. 23 (2016), 493–500.

    Article  MathSciNet  MATH  Google Scholar 

  16. S.-T. Guo, J. Shi, Z.-J. Zhang: Heptavalent symmetric graphs of order 4p. South Asian J. Math. 1 (2011), 131–136. zbl

    MATH  Google Scholar 

  17. X.-H. Hua, Y.-Q. Feng, J. Lee: Pentavalent symmetric graphs of order 2pq. Discrete Math. 311 (2011), 2259–2267.

    Article  MathSciNet  MATH  Google Scholar 

  18. Y. Li, Y.-Q. Feng: Pentavalent one-regular graphs of square-free order. Algebra Colloq. 17 (2010), 515–524.

    Article  MathSciNet  MATH  Google Scholar 

  19. M. W. Liebeck, C. E. Praeger, J. Saxl: A classification of the maximal subgroups of the finite alternating and symmetric groups. J. Algebra 111 (1987), 365–383.

    Article  MathSciNet  MATH  Google Scholar 

  20. P. Lorimer: Vertex-transitive graphs: symmetric graphs of prime valency. J. Graph Theory 8 (1984), 55–68.

    Article  MathSciNet  MATH  Google Scholar 

  21. B. D. McKay: Transitive graphs with fewer than twenty vertices. Math. Comput. 33 (1979), 1101–1121.

    Article  MathSciNet  MATH  Google Scholar 

  22. R. C. Miller: The trivalent symmetric graphs of girth at most six. J. Comb. Theory, Ser. B 10 (1971), 163–182.

    Article  MathSciNet  MATH  Google Scholar 

  23. J.-M. Oh: A classification of cubic s-regular graphs of order 14p. Discrete Math. 309 (2009), 2721–2726.

    Article  MathSciNet  MATH  Google Scholar 

  24. J.-M. Oh: A classification of cubic s-regular graphs of order 16p. Discrete Math. 309 (2009), 3150–3155.

    Article  MathSciNet  MATH  Google Scholar 

  25. J. Pan, B. Ling, S. Ding: One prime-valent symmetric graphs of square-free order. Ars Math. Contemp. 15 (2018), 53–65.

    Article  Google Scholar 

  26. J. Pan, B. Lou, C. Liu: Arc-transitive pentavalent graphs of order 4pq. Electron. J. Comb. 20 (2013), Researh Paper P36, 9 pages.

  27. P. Potocnik: A list of 4-valent 2-arc-transitive graphs and finite faithful amalgams of index (4, 2). Eur. J. Comb. 30 (2009), 1323–1336.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. C. C. Sims: Graphs and finite permutation groups. Math. Z. 95 (1967), 76–86.

    Article  MathSciNet  MATH  Google Scholar 

  30. M. Suzuki: Group Theory I. Grundlehren der Mathematischen Wissenschaften 247, Springer, Berlin, 1982.

    Google Scholar 

  31. R. A. Wilson: The Finite Simple Groups. Graduate Texts in Mathematics 251, Springer, London, 2009.

    Book  Google Scholar 

  32. J. Xu, M. Xu: Arc-transitive Cayley graphs of valency at most four on abelian groups. Southeast Asian Bull. Math. 25 (2001), 355–363.

    Article  MathSciNet  MATH  Google Scholar 

  33. M.-Y. Xu: Automorphism groups and isomorphisms of Cayley digraphs. Discrete Math. 182 (1998), 309–319.

    Article  MathSciNet  MATH  Google Scholar 

  34. J.-X. Zhou, Y.-Q. Feng: Tetravalent s-transitive graphs of order twice a prime power. J. Aust. Math. Soc. 88 (2010), 277–288.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, P.R. China

    Xiao-Hui Hua & Li Chen

Authors
  1. Xiao-Hui Hua
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Li Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiao-Hui Hua.

Additional information

This work was supported by the National Natural Science Foundation of China (11301159, 11671030, 11601132, 11501176, 11526082), the Education Department of Henan Science and Technology Research Key Project (13A110543).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hua, XH., Chen, L. Valency seven symmetric graphs of order 2pq. Czech Math J 68, 581–599 (2018). https://doi.org/10.21136/CMJ.2018.0530-15

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.21136/CMJ.2018.0530-15

Keywords

MSC 2010

Search

Navigation