Skip to main content
SpringerLink
Log in

Heptavalent symmetric graphs of order \(\varvec{24p}\)

  • Published:
Proceedings - Mathematical Sciences Aims and scope Submit manuscript

Abstract

A graph is symmetric if its automorphism group acts transitively on the set of arcs of the graph. In this paper, we classify connected heptavalent symmetric graphs of order 24p for each prime p. As a result, there are twelve sporadic such graphs: one for \(p=2\), four for \(p=3\), one for \(p=5\) and six for \(p=13\).

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. Biggs N, Algebraic graph theory, second ed. (1993) (Cambridge: Cambridge University Press)

    MATH  Google Scholar 

  2. Bondy J A and Murty U S R, Graph theory with applications (1976) (New York: Elsevier Science Ltd)

    Book  Google Scholar 

  3. Bosma W, Cannon C and Playoust C, The MAGMA algebra system I: The user language, J. Symbolic Comput. 24 (1997) 235–265

    Article  MathSciNet  Google Scholar 

  4. Bugeaud Y, Cao Z and Mignotte M, On simple \(K_4\)-groups, J. Algebra 241 (2001) 658–668

    Article  MathSciNet  Google Scholar 

  5. Conway H J, Curtis R T, Norton S P, Parker R A and Wilson R A, Atlas of finite group (1985) (Oxford: Clarendon Press)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Feng Y Q, Kwak J H and Wang K S, Classifying cubic symmetric graphs of order \(8p\) or \(8p^2\), European J. Combin. 26 (2005) 1033–1052

    Article  MathSciNet  Google Scholar 

  8. Gorenstein D, Finite simple groups (1982) (New York: Plenum Press)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  10. Guo S T, Li Y T and Hua X H, \((G,s)\)-transitive graphs of valency \(7\), Algeb. Colloq. 23 (2016) 493–500

    Article  MathSciNet  Google Scholar 

  11. Guo S T, Hou H and Shi J, Pentavalent symmetric graphs of order \(16p\), Acta Math. Appl. Sinica (English Series) 33 (2017) 115–124

    Article  MathSciNet  Google Scholar 

  12. Guo S T, Hou H and Xu Y, Heptavalent symmetric graphs of order \(16p\), Algeb. Colloq. 24 (2017) 453–466

    Article  MathSciNet  Google Scholar 

  13. Huppert B, Eudiche Gruppen I (1967) (Berlin: Springer-Verlag)

  14. Jafarzadeh A and Iranmanesh A, On simple \(K_n\)-groups for \(n=5,6\), in: Groups St. Andrews 2005, edited by C M Campbell, M R Quick, E F Robertson and G C Smith, London Mathematical Society Lecture Note Series, vol. 2 (2007) (Cambridge: Cambridge University Press) pp. 668–680

  15. Li C H, Lu Z P and Wang G X, Arc-transitive graphs of square-free order and small valency, Discrete Math. 339 (2016) 2907–2918

    Article  MathSciNet  Google Scholar 

  16. Ling B, Classifying pentavalnet symmetric graphs of order \(24p\), Bull. Iranian Math. Soc. 43 (2017) 1855–1866

    MathSciNet  MATH  Google Scholar 

  17. Ling B and Lou B, Classifying pentavalent symmetric graphs of order \(40p\), Ars. Comb. 130 (2017) 163–174

    MathSciNet  MATH  Google Scholar 

  18. Ling B, Wu C X and Lou B, Pentavalent symmetric graphs of order \(30p\), Bull. Aust. Math. Soc. 90 (2014) 353–362

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  21. Pan J, Ling B and Ding S, On symmetric graphs of order four times an odd square-free integer and valency seven, Discrete Math. 340 (2017) 2071–2078

    Article  MathSciNet  Google Scholar 

  22. Potočnik P, A list of \(4\)-valent \(2\)-arc-transitive graphs and finite faithful amalgams of index \((4,2)\), European J. Combin. 30 (2009) 1323–1336

    Article  MathSciNet  Google Scholar 

  23. Robinson D J, A course in the theory of groups (1982) (New York: Springer-Verlag)

    Book  Google Scholar 

  24. Sabidussi B O, Vertex-transitive graphs, Monatsh Math. 68 (1964) 426–438

    Article  MathSciNet  Google Scholar 

  25. Shi W J, On simple \(K_4\)-groups, Chinese Science Bull. 36 (1991) 1281–1283 (in Chinese)

    Google Scholar 

  26. Wielandt H, Finite permutation groups (1964) (New York: Academic Press)

    MATH  Google Scholar 

  27. Yang D W and Feng Y Q, Pentavalent symmetric graphs of order \(2p^3\), Sci. China Math. 59 (2016) 1851–1868

    Article  MathSciNet  Google Scholar 

  28. Yang D W, Feng Y Q and Du J L, Pentavalent symmetric graphs of order \(2pqr\), Discrete Math. 339 (2016) 522–532

    Article  MathSciNet  Google Scholar 

  29. Zhou J X, Tetravalent \(s\)-transitive graphs of order \(4p\), Discrete Math. 309 (2009) 6081–6086

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (11301154), and the Innovation Team Funding of Henan University of Science and Technology (2015XTD010).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, 471023, People’s Republic of China

    Song-Tao Guo & Yusen Wu

Authors
  1. Song-Tao Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yusen Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Song-Tao Guo.

Additional information

Communicating Editor: B Sury

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guo, ST., Wu, Y. Heptavalent symmetric graphs of order \(\varvec{24p}\). Proc Math Sci 129, 58 (2019). https://doi.org/10.1007/s12044-019-0497-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12044-019-0497-5

Keywords

2000 Mathematics Subject Classification

Search

Navigation