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On least eigenvalues of bicyclic graphs with fixed number of pendant vertices

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Abstract

We determine the bicyclic graphs of fixed order and number of pendant vertices with exactly two (three, respectively) cycles for which the least eigenvalue is minimal.

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References

  1. F. K. Bell, D. Cvetković, P. Rowlinson, and S. K. Simić, “Graphs for which the least eigenvalue is minimal, I,” Linear Algebra Appl., 429, 234–241 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  2. F. K. Bell, D. Cvetković, P. Rowlinson, and S. K. Simić, “Graphs for which the least eigenvalue is minimal, II,” Linear Algebra Appl., 429, 2168–2179 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  3. D. Cvetković, M. Doob, and H. Sachs, Spectra of Graphs. Theory and Applications, Johann Ambrosius Barth, Heidelberg (1995).

    MATH  Google Scholar 

  4. D. Cvetković and P. Rowlinson, Eigenspace of Graphs, Cambridge Univ. Press, Cambridge (1997).

    Book  Google Scholar 

  5. D. Cvetković, P. Rowlinson, and S. Simić, An Introduction to the Theory of Graph Spectra, Cambridge Univ. Press, Cambridge (2010).

    MATH  Google Scholar 

  6. M. Doob, “A surprising property of the least eigenvalue of a graph,” Linear Algebra Appl., 46, 1–7 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  7. Y. Fan, Y. Wang, and Y. Gao, “Minimizing the least eigenvalues of unicyclic graphs with application to spectral spread,” Linear Algebra Appl., 429, 577–588 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  8. A. J. Hoffman, “On graphs whose least eigenvalue exceeds \( - 1 - \sqrt {2} \),” Linear Algebra Appl., 16, 153–165 (1977).

    Article  MATH  Google Scholar 

  9. A. J. Hoffman and J. H. Smith, in: Recent Advances in Graph Theory, Academia Praha, New York (1975), pp. 273–281.

    Google Scholar 

  10. Y. Hong, “Bounds of eigenvalues of a graph,” Acta Math. Appl. Sinica, 4, 165–168 (1988).

    Article  MATH  Google Scholar 

  11. Y. Hong and J. Shu, “Sharp lower bounds of the least eigenvalue of planar graphs,” Linear Algebra Appl., 296, 227–232 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  12. R. Liu, M. Zhai, and J. Shu, “The least eigenvalue of unicyclic graphs with n vertices and k pendant vertices,” Linear Algebra Appl., 431, 657–665 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  13. Q. Li, K. Feng, “On the largest eigenvalues of graphs,” Acta Math. Appl. Sinica, 2 167–175 (1979) (in Chinese).

    MathSciNet  Google Scholar 

  14. M. Petrović, B. Borovićanin, and T. Aleksić, “Bicyclic graphs for which the least eigenvalue is minimum,” Linear Algebra Appl., 430, 1328–1335 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  15. S. Simić, “On the largest eigenvalue of unicyclic graphs,” Publ. Inst. Math. Beograd, 42, 13–19 (1987).

    MathSciNet  Google Scholar 

  16. Y. Tan and Y. Fan, “The vertex (edge) independence number, vertex (edge) cover number and the least eigenvalue of a graph,” Linear Algebra Appl., 433, 790–795 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  17. M. Ye, Y. Fan, and D. Liang, “The least eigenvalue of graphs with given connectivity,” Linear Algebra Appl., 430, 1375–1379 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  18. Y. Wang, Y. Fan, “The least eigenvalue of a graph with cut vertices,” Linear Algebra Appl., 433, 19–27 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  19. B. Wu, E. Xiao, Y. Hong, “The spectral radius of trees on k pendant vertices,” Linear Algebra Appl., 395, 343–349 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  20. M. Zhai, R. Liu, and J. Shu, “Minimizing the least eigenvalue of unicyclic graphs with fixed diameter,” Discr. Math., 310, 947–955 (2010).

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, South China Normal University, Guangzhou, P. R. China

    Zhongzhu Liu & Bo Zhou

Authors
  1. Zhongzhu Liu
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  2. Bo Zhou
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Correspondence to Zhongzhu Liu.

Additional information

Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 71, Algebraic Techniques in Graph Theory and Optimization, 2011.

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Liu, Z., Zhou, B. On least eigenvalues of bicyclic graphs with fixed number of pendant vertices. J Math Sci 182, 175–192 (2012). https://doi.org/10.1007/s10958-012-0738-y

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  • DOI: https://doi.org/10.1007/s10958-012-0738-y

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