Advertisement

Journal of Algebraic Combinatorics

, Volume 47, Issue 4, pp 585–601 | Cite as

Integral Cayley graphs over dihedral groups

Article

Abstract

In this paper, we give a necessary and sufficient condition for the integrality of Cayley graphs over the dihedral group \(D_n=\langle a,b\mid a^n=b^2=1,bab=a^{-1}\rangle \). Moreover, we also obtain some simple sufficient conditions for the integrality of Cayley graphs over \(D_n\) in terms of the Boolean algebra of \(\langle a\rangle \), from which we find infinite classes of integral Cayley graphs over \(D_n\). In particular, we completely determine all integral Cayley graphs over the dihedral group \(D_p\) for a prime p.

Keywords

Cayley graph Integral graph Dihedral group Character 

Mathematics Subject Classification

05C50 

Notes

Acknowledgements

The authors would like to thank the referees and the editors for very careful reading and for helpful comments which helped to improve the presentation of the paper.

References

  1. 1.
    Abdollahi, A., Vatandoost, E.: Which Cayley graphs are integral?, Electron. J. Combin. 16 #R122 (2009)Google Scholar
  2. 2.
    Alperin, R.C.: Rational subsets of finite groups. Int. J. Group Theory 2, 53–55 (2014)MathSciNetMATHGoogle Scholar
  3. 3.
    Alperin, R.C., Peterson, B.L.: Integral sets and Cayley graphs of finite groups. Electron. J. Comb. 19, P44 (2012)MathSciNetMATHGoogle Scholar
  4. 4.
    Babai, L.: Spectra of Cayley graphs. J. Combin. Theory Ser. B 27, 180–189 (1979)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bridges, W.G., Mena, R.A.: Rational \(G\)-matrices with rational eigenvalues. J. Combin. Theory Ser. A 32, 264–280 (1982)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Brouwer, A.E.: Small integral trees. Electron. J. Combin. 15, #N1 (2008)Google Scholar
  7. 7.
    Bussemaker, F.C., Cvetković, D.: There are exactly \(13\) connected, cubic, integral graphs, Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. Fiz. 544–576:43–48 (1976)Google Scholar
  8. 8.
    Cheng, Y.K., Lau, T., Wong, K.B.: Cayley graph on symmetric group generated by elements fixing \(k\) points. Linear Algebra Appl. 471, 405–426 (2014)MathSciNetMATHGoogle Scholar
  9. 9.
    Csikvári, P.: Integral trees of arbitrarily large diameters. J. Algebr. Comb. 32, 371–377 (2010)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Davidoff, G., Sarnak, P., Valette, A.: Elementary Number Theory, Group Theory and Ramanujan Graphs. Cambridge University Press, New York (2003)CrossRefMATHGoogle Scholar
  11. 11.
    DeVos, M., Krakovski, R., Mohar, B., Ahmady, A.S.: Integral Cayley multigraphs over Abelian and Hamiltonian groups. Electron. J. Comb. 20, P63 (2013)MathSciNetMATHGoogle Scholar
  12. 12.
    Harary, F., Schwenk, A.J.: Which graphs have integral spectra? in Graphs and Combinatorics. Lecture Notes in Math, vol. 406. Springer, Berlin (1974)Google Scholar
  13. 13.
    Jedwab, J., Davis, J.: A survey of Hadamard difference sets, A Special Research Quarter on Groups, Difference Sets, and the Monster Walter de Gruyter and Co. pp. 145–156 (1996)Google Scholar
  14. 14.
    Klotz, W., Sander, T.: Integral Cayley graphs over abelian groups. Electron. J. Comb. 17, R81 (2010)MathSciNetMATHGoogle Scholar
  15. 15.
    Lander, E.S.: Symmetric Designs: an Algebraic Approach, London Mathematical Society Lecture Notes Series 74. Cambridge University Press, Cambridge (1983)CrossRefGoogle Scholar
  16. 16.
    Lepović, M. , Simić, S.K., Balińska, K.T., Zwierzyński, K.T.: There are 93 non-regular, bipartite integral graphs with maximum degree four, The Technical University of Poznań, CSC Report 511 (2005)Google Scholar
  17. 17.
    Serre, J.P.: Linear Representations of Finite Groups, Springer, New York, Translated from the second French edition by L. Scott, Graduate Texts in Mathematics, Vol. 42 (1997)Google Scholar
  18. 18.
    So, W.: Integral circulant graphs. Discrete Math. 306, 153–158 (2005)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Stevanović, D.: 4-Regular integral graphs avoiding \(\pm 3\) in the spectrum, Univ. Beograd. Publ. Elektrotehn. Fak., Ser. Mat. 14: 99–110 (2003)Google Scholar
  20. 20.
    Wang, L., Li, X., Liu, R.: Integral trees with diameter 6 or 8. Electron. Notes Discrete Math. 3, 208–212 (1999)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Watanabe, M.: Note on integral trees. Math. Rep. Toyama Univ. 2, 95–100 (1979)MathSciNetMATHGoogle Scholar
  22. 22.
    Watanabe, M., Schwenk, A.J.: Integral starlike trees. J. Austral Math. Soc. 28, 120–128 (1979)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Zhang, D., Tan, S.: On integral trees of diameter 4 (in Chinese). J. Syst. Sci. Math. Sci. 20 (2000)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.College of Mathematics and Systems ScienceXinjiang UniversityÜrümqiPeople’s Republic of China

Personalised recommendations