Advertisement

The spectra of lifted digraphs

  • C. Dalfó
  • M. A. FiolEmail author
  • J. Širáň
Article

Abstract

We present a method to derive the complete spectrum of the lift \(\varGamma ^\alpha \) of a base digraph \(\varGamma \), with voltage assignment \(\alpha \) on a (finite) group G. The method is based on assigning to \(\varGamma \) a quotient-like matrix whose entries are elements of the group algebra \(\mathbb {C}[G]\), which fully represents \(\varGamma ^{\alpha }\). This allows us to derive the eigenvectors and eigenvalues of the lift in terms of those of the base digraph and the irreducible characters of G. Thus, our main theorem generalizes some previous results of Lovász and Babai concerning the spectra of Cayley digraphs.

Keywords

Digraph Adjacency matrix Regular partition Quotient digraph Spectrum Lifted digraph 

Mathematics Subject Classification

05C20 05C50 15A18 

Notes

References

  1. 1.
    Babai, L.: Spectra of Cayley graphs. J. Combin. Theory Ser. B 27, 180–189 (1979)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bang-Jensen, J., Gutin, G.: Digraphs. Theory, Algorithms and Applications. Springer Monographs in Mathematics, 2nd edn. Springer, London (2009)zbMATHGoogle Scholar
  3. 3.
    Burrow, M.: Representation Theory of Finte Groups. Dover Publications, Inc., New York (1993)Google Scholar
  4. 4.
    Cvetković, D., Doob, M., Sachs, H.: Spectra of Graphs. Theory and Applications, 3rd edn. Johann Ambrosius Barth, Heidelberg (1995)zbMATHGoogle Scholar
  5. 5.
    Dalfó, C., Fiol, M.A., Miller, M., Ryan, J., Širáň, J.: An Algebraic Approach to Lifts of Digraphs. Discrete Appl. Math. (2018, to appear)Google Scholar
  6. 6.
    Diestel, R.: Graph Theory, Graduate Texts in Mathematics, vol. 173, 4th edn. Springer, Heilderberg (2010)Google Scholar
  7. 7.
    Feng, R., Kwak, J.H., Lee, J.: Characteristic polynomials of graph coverings. Bull. Aust. Math. Soc. 69, 133–136 (2004)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Gera, R., Stǎnicǎ, P.: The spectrum of generalized Petersen graphs. Australas. J. Combin. 49, 39–45 (2011)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Godsil, C.D., Hensel, A.D.: Distance regular covers of the complete graph. J. Combin. Theory Ser. B 56(2), 205–238 (1992)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Gould, H.W.: The Girard-Waring power sum formulas for symmetric functions and Fibonacci sequences. Fibonacci Quart. 37(2), 135–140 (1999)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Gross, J.L., Tucker, T.W.: Generating all graph coverings by permutation voltage assignments. Discrete Math. 18, 273–283 (1977)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Klin, M., Pech, C.: A new construction of antipodal distance regular covers of complete graphs through the use of Godsil–Hensel matrices. Ars Math. Contemp. 4(2), 205–243 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kwak, J.H., Lee, J.: Characteristic polynomials of some graph bundles II. Linear Multilinear Algebra 32, 61–73 (1992)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kwak, J.H., Kwon, Y.S.: Characteristic polynomials of graph bundles having voltages in a dihedral group. Linear Algebra Appl. 336, 99–118 (2001)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Lovász, L.: Spectra of graphs with transitive groups. Period. Math. Hungar. 6, 191–196 (1975)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Miller, M., Širáň, J.: Moore graphs and beyond: a survey of the degree/diameter problem. Electron. J. Combin. 20(2), DS14v2 (2013)Google Scholar
  17. 17.
    Mizuno, H., Sato, I.: Characteristic polynomials of some graph coverings. Discrete Math. 142, 295–298 (1995)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Departament de MatemàticaUniversitat de LleidaIgualadaSpain
  2. 2.Departament de Matemàtiques, Barcelona Graduate SchoolUniversitat Politècnica de CatalunyaBarcelonaSpain
  3. 3.Department of Mathematics and StatisticsThe Open UniversityMilton KeynesUK
  4. 4.Department of Mathematics and Descriptive GeometrySlovak University of TechnologyBratislavaSlovak Republic

Personalised recommendations