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Ihara Zeta Function and Spectrum of the Cone Over a Semiregular Bipartite Graph

  • Deqiong Li
  • Yaoping HouEmail author
Original Paper
  • 10 Downloads

Abstract

In this paper, a formula for the Ihara zeta function of the cone over a semiregular bipartite graph is derived. Using this formula, we show that two cones over semiregular bipartite graphs are cospectral if and only if they have the same Ihara zeta function. Moreover, the convergence of the zeta function of this family of graphs is considered.

Keywords

Ihara zeta function Cone over semiregular bipartite graph Spectrum Pole Complexity 

Mathematics Subject Classification

05C50 15A15 

Notes

Acknowledgements

This work is supported by the National Natural Science Foundation of China (No. 11571101) and the Natural Science Foundation of Hunan Province, China (Grant 2019JJ40184, 2019JJ50173). The authors would like to thank the anonymous referees for their valuable comments and careful reviewing of this paper. In particular, the suggestion on the simplication and improvement for the proof of Theorem 6 is gratefully acknowledged.

References

  1. 1.
    Bayati, P., Somodi, M.: On the Ihara zeta function of cones over regular graphs. Graphs Combin. 29, 1633–1646 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bass, H.: The Ihara–Selberg zeta function of a tree lattice. Int. J. Math. 3, 717–797 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Blanchard, A., Pakala, E., Somodi, M.: Ihara zeta function and cospectrality of joins of regular graphs. Discr. Math. 333, 84–93 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cooper, Y.: Properties determined by the Ihara zeta function of a graph. Electron. J. Combin. 16, R84 (2009)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Cvetković, D., Doob, M., Sachs, H.: Spectra of Graphs: Theory and Application. Johann Ambrosius Barth Verlag, Heidelberg (1995)zbMATHGoogle Scholar
  6. 6.
    Czarneski, D.: Zeta functions of finite graphs (Ph. D. dissertation), LSU (2005)Google Scholar
  7. 7.
    Durfee, C., Martin, K.: Distinguishing graphs with zeta functions and generalized spectra. Linear Algebra Appl. 481, 54–82 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hashimoto, K.: Zeta functions of finite graphs and representations of \(p\)-Adic Groups. Adv. Stud. Pure Math. 15, 211–280 (1989)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kotani, M., Sunada, T.: Zeta functions of finite graphs. J. Math. Sci. 7(1), 7–25 (2000)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Northshield, S.: A note on the zeta function of a graph. J. Combin. Theory Ser. B 74, 408–410 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Sato, I.: Zeta functions and complexities of a semiregular bipartite graph and its line graph. Discr. Math. 307, 237–245 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Sato, I.: Zeta functions and complexities of middle graphs of semiregular bipartite graphs. Discr. Math. 335, 92–99 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Setyadi, A., Storm, C.K.: Enumeration of graphs with the same Ihara zeta function. Linear Algebra Appl. 438, 564–572 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Terras, A.: Zeta Funtions of Graphs: A Stroll Through the Garden. Cambridge University Press, Cambridge (2011)zbMATHGoogle Scholar
  15. 15.
    Zhang, F.Z.: The Schur Complement and Its Applications. Springer, New York (2005)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Key Laboratory of High Performance Computing and Stochastic Information Processing, College of Mathematics and StatisticsHunan Normal UniversityChangshaChina
  2. 2.College of Mathematics and ComputationHunan University of Science and TechnologyXiangtanChina

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