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Maxima of the Q-Spectral Radius of \(C_3\) (\(C_4\))-Free Graphs with Given Size and Minimum Degree \(\delta \ge 2\)

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Abstract

In this paper, we study the signless Laplacian spectral Turán type problem in terms of the size, obtain sharp upper bounds on the signless spectral radius of \(C_3\) (\(C_4\))-free graphs with given size and minimum degree \(\delta \ge 2\), and characterize the corresponding extremal graphs completely.

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References

  1. Brualdi, R.A., Hoffman, A.J.: On the spectral radius of \((0,1)\)-matrices. Linear Algebra Appl. 65, 133–146 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen, W., Wang, B., Zhai, M.: Signless Laplacian spectral radius of graphs without short cycles or long cycles. Linear Algebra Appl. 645, 123–136 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cvetković, D., Rowlinson, P., Simić, S.K.: Eigenvalue bounds for the signless Laplacian. Publ. Inst. Math. (Beograd) 81(95), 11–27 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cvetković, D., Simić, S.: Towards a spectral theory of graphs based on the signless Laplacian, I. Publ. Inst. Math. (Beograd) 85(99), 19–33 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Das, K.C.: The Laplacian spectrum of a graph. Comput. Appl. Math. 48, 715–724 (2004)

    MathSciNet  MATH  Google Scholar 

  6. Feng, L., Yu, G.: On three conjectures involving the signless Laplacian spectral radius of graphs. Publ. Inst. Math. (Beograd) 85(99), 35–38 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gao, M., Lou, Z., Huang, Q.: A sharp upper bound on the spectral radius of \(C_5\)-free /\(C_6\)-free graphs with given size. Linear Algebra Appl. 640, 162–178 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  8. Guo, S.G., Zhang, R.: Sharp upper bounds on the \(Q\)-index of (minimally) \(2\)-connected graphs with given size. Discrete Appl. Math. 320, 408–415 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hong, Y., Zhang, X.D.: Sharp upper and lower bounds for largest eigenvalue of the Laplacian matrices of trees. Discrete Math. 296, 187–197 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Jia, H., Li, S., Wang, S.: Ordering the maxima of \(L\)-index and \(Q\)-index: graphs with given size and diameter. Linear Algebra Appl. 652, 18–36 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  11. Li, J.S., Zhang, X.D.: On the Laplacian eigenvalues of a graph. Linear Algebra Appl. 285, 305–307 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lin, H., Ning, B., Wu, B.Y.D.R.: Eigenvalues and triangles in graphs. Combin. Probab. Comput. 30, 258–270 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lou, Z., Guo, J.M., Wang, Z.: Maxima of \(L\)-index and \(Q\)-index: graphs with given size and diameter. Discrete Math. 344, 112533 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  14. Merris, R.: A note on Laplacian graph eigenvalues. Linear Algebra Appl. 285, 33–35 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. Nikiforov, V.: Some inequalities for the largest eigenvalue of a graph. Combin. Probab. Comput. 11, 179–189 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Nikiforov, V.: The maximum spectral radius of \(C_4\)-free graphs of given order and size. Linear Algebra Appl. 430, 2898–2905 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Nikiforov, V.: The spectral radius of graphs without paths and cycles of specified length. Linear Algebra Appl. 432, 2243–2256 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Nosal, E.: Eigenvalues of Graphs, Master Thesis, University of Calgary (1970)

  19. Oliveira, C.S., de Limab, L.S., de Abreu, N.M.M., Hansen, P.: Bounds on the index of the signless Laplacian of a graph. Discrete Appl. Math. 158, 355–360 (2010)

    Article  MathSciNet  Google Scholar 

  20. Rowlinson, P.: On the maximal index of graphs with a prescribed number of edges. Linear Algebra Appl. 110, 43–53 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wang, Z., Guo, J.M.: Maximum degree and spectral radius of graphs in terms of size. arXiv:2208.13139

  22. Yu, G.L., Wu, Y.R., Shu, J.L.: Sharp bounds on the signless Laplacian spectral radii of graphs. Linear Algebra Appl. 603, 683–687 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhai, M., Lin, H., Shu, J.: Spectral extrema of graphs with fixed size: cycles and complete bipartite graphs. Eur. J. Combin. 95, 103322 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  24. Zhai, M.Q., Xue, J., Liu, R.: An extremal problem on \(Q\)-spectral radii of graphs with given size and matching number. Linear Multilinear Algebra 70, 5334–5345 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhai, M., Xue, J., Lou, Z.: The signless Laplacian spectral radius of graphs with a prescribed number of edges. Linear Algebra Appl. 603, 154–165 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  26. Zhang, R., Guo, S.G.: Ordering graphs with given gize by their signless Laplacian spectral radii. Bull. Malays. Math. Sci. Soc. 45, 2165–2174 (2022)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors are grateful to the referees for their careful reading and many valuable suggestions.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Yancheng Teachers University, Yancheng, 224002, Jiangsu, People’s Republic of China

    Shu-Guang Guo & Rong Zhang

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  1. Shu-Guang Guo
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  2. Rong Zhang
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Correspondence to Shu-Guang Guo.

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Supported by the National Natural Science Foundation of China (nos. 12071411, 12171222).

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Guo, SG., Zhang, R. Maxima of the Q-Spectral Radius of \(C_3\) (\(C_4\))-Free Graphs with Given Size and Minimum Degree \(\delta \ge 2\). Graphs and Combinatorics 39, 102 (2023). https://doi.org/10.1007/s00373-023-02685-1

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  • DOI: https://doi.org/10.1007/s00373-023-02685-1

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