Tetravalent edge-transitive Cayley graphs of Frobenius groups

Journal of Algebraic Combinatorics (2021)Cite this article

Abstract

In this paper, we give a characterisation for a class of edge-transitive Cayley graphs and provide a method for constructing edge-transitive graphs of valency 4 with arbitrarily large vertex stabiliser. In particular, in the last section, we obtain certain extensions of the results of Li et al. (Tetravalent edge-transitive Cayley graphs with odd number of vertices, J Comb Theory Ser B 96:164–181, 2006) on half-transitive graphs.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

US$ 39.95

Tax calculation will be finalised during checkout.

References

  1. 1.

    Al-bar, J.A., Al-kenani, A.N., Muthana, N.M., Praeger, C.E., Spiga, P.: Finite edge-transitive oriented graphs of valency four: a global approach. Electron. J. Comb. 23, 1–10 (2016)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Baik, Y.G., Feng, Y.Q., Sim, H.S., Xu, M.Y.: On the normality of Cayley graphs of abelian groups. Algebra Colloq. 5, 297–304 (1998)

    MathSciNet  MATH  Google Scholar 

  3. 3.

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

    MathSciNet  Article  Google Scholar 

  4. 4.

    Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, P.: Atlas of Finite Groups. Oxford University Press, Oxford (1985)

    Google Scholar 

  5. 5.

    Corr, B.P., Praeger, C.E.: Normal edge-transitive Cayley graphs of Frobenius groups. J. Algebr. Comb. 42, 803–827 (2015)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Detinko, A.S., Flannery, D.L.: Nilpotent primitive linear groups over finite fields. Commun. Algebra. 33, 497–505 (2005)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Dixon, J.D., Mortimer, B.: Permutation Groups. Springer, New York (1996)

    Google Scholar 

  8. 8.

    Doerk, K., Hawkes, T.: Finite Soluble Groups. Walter de Gruyter Co., Berlin (1992)

    Google Scholar 

  9. 9.

    Du, S.F., Xu, W.Q.: \(2\)-arc-transitive regular covers of \({ K}_{n, n}-n{ K}_2\) having the covering transformation group \({\mathbb{Z}}_p^3\). J. Aust. Math. Soc. 101, 145–170 (2016)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Fang, X.G., Li, C.H., Xu, M.Y.: On edge-transitive Cayley graphs of valency four. Eur. J. Comb. 25, 1107–1116 (2004)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Feng, Y.Q., Kwak, J.H., Wang, X.Y., Zhou, J.X.: Tetravalent half-arc-transitive graphs of order \(2pq\). J. Algebr. Comb. 33, 543–553 (2011)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Feng, Y.Q., Kwak, J.H., Xu, M.Y., Zhou, J.X.: Tetravalent half-arc-transitive graphs of order \(p^4\). Eur. J. Comb. 29, 555–567 (2008)

    Article  Google Scholar 

  13. 13.

    Godsil, C.D.: On the full automorphism group of a graph. Combinatorica 1, 243–256 (1981)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Gorenstein, D.: Finite Groups. Harper and Row, New York (1968)

    Google Scholar 

  15. 15.

    Gorenstein, D.: Finite Simple Groups. Plenum Press, New York (1982)

    Google Scholar 

  16. 16.

    Guralnick, R.: Subgroups of prime power index in a simple group. J. Algebra 81, 304–311 (1983)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Kazarin, L.S.: Groups that can be represented as a product of two solvable subgroups. Commun. Algebra 14, 1001–1066 (1986)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Kutnar, K., Marušič, D., Sp̌arl, P.: An infinite family of half-arc-transitive graphs with universal reachability relation. Eur. J. Comb. 31, 1725–1734 (2010)

  19. 19.

    Kuzman, B.: Arc-transitive elementary abelian covers of the complete graph \( K_5\). Linear Algebra Appl. 433, 1909–1921 (2010)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Li, C.H.: Finite \(s\)-arc transitive Cayley graphs and flag-transitive projective planes. Proc. Amer. Math. Soc. 133, 31–41 (2005)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Li, C.H., Liu, Z., Lu, Z.P.: The edge-transitive tetravalent Cayley graphs of square-free order. Discrete Math. 312, 1952–1967 (2012)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Li, C.H., Lu, Z.P., Zhang, H.: Tetravalent edge-transitive Cayley graphs with odd number of vertices. J. Comb. Theory Ser. B. 96, 164–181 (2006)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Li, C.H., Pan, J.M., Song, S.J., Wang, D.J.: A characterization of a family of edge-transitive metacirculant graphs. J. Comb. Theory Ser. B. 107, 12–25 (2014)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Li, C.H., Rao, G., Song, S.J.: On finite self-complementary metacirculants. J. Algebr. Comb. 40, 1135–1144 (2014)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Li, C.H., Xia, B.Z.: Factorizations of almost simple groups with a solvable factor, and Cayley graphs of solvable groups, Mem. Amer. Math. Soc. (2020, in press)

  26. 26.

    Liu, H.L., Wang, L.: Cubic arc-transitive Cayley graphs on Frobenius groups. J. Algebra Appl. 17, 1–9 (2018)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Marušič, D., Nedela, R.: Maps and half-transitive graphs of valency \(4\). Eur. J. Comb. 19, 345–354 (1998)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Marušič, D., Sp̌arl, P.: On quartic half-arc-transitive metacirculants. J. Algebr. Comb. 28, 365–395 (2008)

  29. 29.

    Pan, J.M., Liu, Y., Huang, Z.H., Liu, C.L.: Tetravalent edge-transitive graphs of order \(p^2q\). Sci. China Math. 57(2), 293–302 (2014)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Praeger, C.E.: An O’Nan-Scott theorem for finite quasiprimitive permutation groups and an application to \(2\)-arc transitive graphs. J. Lond. Math. Soc. 47, 227–239 (1992)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Praeger, C.E.: Finite normal edge-transitive Cayley graphs. Bull. Aust. Math. Soc. 60(2), 207–220 (1999)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Song, S.J., Li, C.H., Wang, D.J.: Classifying a family of edge-transitive metacirculant graphs. J. Algebr. Comb. 35, 497–513 (2012)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Song, S.J., Li, C.H., Wang, D.J.: A family of edge-transitive Frobenius metacirculants of small valency. Eur. J. Comb. 34, 512–521 (2013)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Suzuki, M.: Group Theory I. Springer, Berlin, New York (1982)

    Google Scholar 

  35. 35.

    Wang, X.Y., Feng, Y.Q.: Tetravalent half-edge-transitive graphs and non-normal Cayley graphs. J. Graph Theory 70(2), 197–213 (2012)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Weiss, R.M.: \(s\)-transitive graphs. Algebr. Methods Graph Theory 2, 827–847 (1981)

    MATH  Google Scholar 

  37. 37.

    Xu, M.Y.: Half-transitive graphs of prime-cube order. J. Algebr. Comb. 1(3), 275–282 (1992)

    MathSciNet  Article  Google Scholar 

  38. 38.

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

    MathSciNet  Article  Google Scholar 

  39. 39.

    Xu, W.Q., Zhu, Y.H., Du, S.F.: \(2\)-arc-transitive regular covers of \({ K}_{n, n}-n{ K}_2\) with the covering transformation group \({\mathbb{Z}}_p^2\). Ars Math. Contemp. 10, 269–280 (2016)

    MathSciNet  Article  Google Scholar 

  40. 40.

    Zhou, C.X., Feng, Y.Q.: An infinite family of tetravalent half-arc-transitive graphs. Discrete Math. 306, 2205–2211 (2006)

    MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

  1. School of Mathematics and Statistics, Yunnan University, Kunming, 650091, Yunnan, People’s Republic of China

    Lei Wang

  2. Department of Mathematics, Yunnan Normal University, Kunming, 650500, Yunnan, People’s Republic of China

    Yin Liu

  3. School of Mathematics and Statistics, Southwest University, Beibei, Chongqing, 400715, People’s Republic of China

    Yanxiong Yan

Authors
  1. Lei Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yin Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yanxiong Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yin Liu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported by Natural Science Foundation of China (No. 12061083); Educational Department Fund of Yunnan (No. 2019J0026); NSF of Yunnan Province (No. 2017FD071); Natural Science Foundation of China (Nos. 11671324; 11971391); Fundamental Research Funds for the Central Universities (Nos. XDJK2019C116; XDJK2019B030) and Teaching Reform Project of Southwest University (No. 2018JY061).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wang, L., Liu, Y. & Yan, Y. Tetravalent edge-transitive Cayley graphs of Frobenius groups. J Algebr Comb (2021). https://doi.org/10.1007/s10801-020-01005-7

Download citation

Keywords

  • Frobenius group
  • Edge-transitive graph
  • Coset graph
  • Cayley graph

Mathematics Subject Classification

  • 05C25
  • 05E18