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Laplacian of a Graph Covering and Its Applications

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Abstract

Let G be a finite graph and \(\overline{G}\) be any graph covering over G. By applying the representation theory of symmetric groups, the Laplacian characteristic polynomial and the normalized Laplacian characteristic polynomial of \(\overline{G}\) are investigated. As applications, adopting the algebra method the Kirchhoff index, the multiplicative degree-Kirchhoff index and the complexity of any connected covering over a connected graph are derived.

Keywords

Graph covering Laplacian matrix Normalized Laplacian matrix Resistance distance Kirchhoff index Complexity 

Mathematics Subject Classification

05C50 

Notes

Acknowledgements

This project was supported by the National Natural Science Foundation of China (No. 11571101). The authors are grateful to the anonymous referees for their valuable comments and helpful suggestions, which have considerably improved the presentation of this paper.

References

  1. 1.
    Bapat, R.B.: Graph and Matrices. Universitext. Springer/Hindustan Book Agency, London/New Delhi (2010)CrossRefGoogle Scholar
  2. 2.
    Bapat, R.B., Gutman, I., Xiao, W.J.: A simple method for computing resistance distance. Z. Naturforch 58a, 494–498 (2003)Google Scholar
  3. 3.
    Chen, H.Y., Zhang, F.J.: Resistance distance and the normalized Laplacian spectrum. Discrete Appl. Math. 155, 654–661 (2007)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Cvetkovič, D., Rowlinson, P., Simić, S.: An Introduction to the Theory of Graph Spectra. London Mathematical Society Student Texts. Cambridge University, London (2010)MATHGoogle Scholar
  5. 5.
    Deng, A.P., Sato, I., Wu, Y.K.: Characteristic polynomials of ramified uniform covering digraphs. Eur. J. Comb. 28(4), 1099–1114 (2007)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Feng, R.Q., Kwak, J.H., Lee, J.: Characteristic polynomials of graph coverings. Bull. Aust. Math. Soc. 69, 133–136 (2004)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Feng, L.H., Yu, G.H., Liu, W.J.: Further result regarding the degree Kirchhoff index of graphs. Miskolc Math. Notes 15(1), 97–108 (2014)MathSciNetMATHGoogle Scholar
  8. 8.
    Gross, J.L., Tucker, T.W.: Generating all graph coverings by permutation voltage assignments. Discrete Math. 18, 273–283 (1977)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Gutman, I., Mohar, B.: The quasi-Wiener and the Kirchhoff indices coincide. J. Chem. Inf. Comput. Sci. 36, 982–985 (1996)CrossRefGoogle Scholar
  10. 10.
    Harary, F.: On the notion of balanced in a signed graph. Mich. Math. J. 2(1), 143–146 (1953)MATHGoogle Scholar
  11. 11.
    Hou, Y.P., Li, J.S., Pan, Y.L.: On the Laplacian eigenvalues of signed graphs. Linear Multilinear Algebra 51(1), 21–30 (2003)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Huang, Q.Y., Chen, H.Y., Deng, Q.Y.: Resistance distances and the Kirchhoff index in double graphs. J. Appl. Math. Comput. 50(1), 1–14 (2014)MathSciNetMATHGoogle Scholar
  13. 13.
    Klein, D.J., Randić, M.: Resistance distance. J. Math. Chem. 12, 81–95 (1993)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kwak, J.H., Lee, J.: Characteristic polynomials of some graph bundles II. Linear Multilinear Algebra 32, 61–73 (1992)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Mizuno, H., Sato, I.: Characteristic polynomials of some graph coverings. Discrete Math. 142, 295–298 (1995)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Serre, J.P.: Linear Representations of Finite Group. Springer, New York (1977)CrossRefGoogle Scholar
  17. 17.
    Shi, L.Y., Chen, H.Y.: Resistance distance and Kirchhoff index of graphs with an involution. Discrete Appl. Math. 215, 185–196 (2016)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Stark, H.M., Terras, A.A.: Zeta functions of finite graphs and coverings. Adv. Math. 121, 124–165 (1996)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Yang, Y.J., Klein, D.J.: Resistance distance-based graph invariants of subdivisions and triangulations of graphs. Discrete Appl. Math. 181, 260–274 (2015)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Zaslavsky, T.: Signed graphs. Discrete Appl. Math. 4, 47–74 (1982)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2018

Authors and Affiliations

  1. 1.Key Laboratory of HPCSIP, Ministry of Education of China, College of Mathematics and Computer ScienceHunan Normal UniversityChangshaChina
  2. 2.College of Mathematics and ComputationHunan Science and Technology UniversityXiangtanChina

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