Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

A𝛼-SPECTRAL EXTREMA OF GRAPHS WITH GIVEN SIZE AND MATCHING NUMBER

  • Xingyu Lei (Faculty of Mathematics and Statistics Central China Normal University) ;
  • Shuchao Li (Faculty of Mathematics and Statistics Central China Normal University) ;
  • Jianfeng Wang (School of Mathematics and Statistics Shandong University of Technology)
  • Received : 2022.05.15
  • Accepted : 2023.04.21
  • Published : 2023.07.31
Download PDF
⟨ Previous Next ⟩

Abstract

In 2017, Nikiforov proposed the A𝛼-matrix of a graph G. This novel matrix is defined as A𝛼(G) = 𝛼D(G) + (1 - 𝛼)A(G), 𝛼 ∈ [0, 1], where D(G) and A(G) are the degree diagonal matrix and adjacency matrix of G, respectively. Recently, Zhai, Xue and Liu [39] considered the Brualdi-Hoffman-type problem for Q-spectra of graphs with given matching number. As a continuance of it, in this contribution we consider the Brualdi-Hoffman-type problem for A𝛼-spectra of graphs with given matching number. We identify the graphs with given size and matching number having the largest A𝛼-spectral radius for ${\alpha}{\in}[{\frac{1}{2}},1)$.

Keywords

References

  1. R. B. Bapat, Graphs and Matrices, Universitext, Springer, London, 2010. https://doi.org/10.1007/978-1-84882-981-7
  2. F. Belardo, M. Brunetti, and A. Ciampella, On the multiplicity of α as an Aα(Γ)-eigenvalue of signed graphs with pendant vertices, Discrete Math. 342 (2019), no. 8, 2223-2233. https://doi.org/10.1016/j.disc.2019.04.024
  3. A. E. Brouwer and W. H. Haemers, Spectra of Graphs, Universitext, Springer, New York, 2012. https://doi.org/10.1007/978-1-4614-1939-6
  4. R. A. Brualdi and A. J. Hoffman, On the spectral radius of (0, 1)-matrices, Linear Algebra Appl. 65 (1985), 133-146. https://doi.org/10.1016/0024-3795(85)90092-8
  5. D. M. Cardoso, G. Pasten, and O. L. Rojo, On the multiplicity of α as an eigenvalue of Aα(G) of graphs with pendant vertices, Linear Algebra Appl. 552 (2018), 52-70. https://doi.org/10.1016/j.laa.2018.04.013
  6. Q. Chen and Q. Huang, The maximal Aα-spectral radius of graphs with given matching number, Linear Multilinear Algebra 70 (2022), no. 20, 5193-5206. https://doi.org/10.1080/03081087.2021.1910120
  7. Y. Y. Chen, D. Li, and J. Meng, On the second largest Aα-eigenvalues of graphs, Linear Algebra Appl. 580 (2019), 343-358. https://doi.org/10.1016/j.laa.2019.06.027
  8. Y. Y. Chen, D. Li, Z. Wang, and J. Meng, Aα-spectral radius of the second power of a graph, Appl. Math. Comput. 359 (2019), 418-425. https://doi.org/10.1016/j.amc.2019.04.077
  9. Z. Feng and W. Wei, On the Aα-spectral radius of graphs with given size and diameter, Linear Algebra Appl. 650 (2022), 132-149. https://doi.org/10.1016/j.laa.2022.06.006
  10. L. H. Feng, G. Yu, and X. D. Zhang, Spectral radius of graphs with given matching number, Linear Algebra Appl. 422 (2007), no. 1, 133-138. https://doi.org/10.1016/j.laa.2006.09.014
  11. S. Friedland, The maximal eigenvalue of 0-1 matrices with prescribed number of ones, Linear Algebra Appl. 69 (1985), 33-69. https://doi.org/10.1016/0024-3795(85)90068-0
  12. R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge Univ. Press, Cambridge, 1985. https://doi.org/10.1017/CBO9780511810817
  13. P. Huang, J. X. Li, and W. C. Shiu, Maximizing the Aα-spectral radius of graphs with given size and diameter, Linear Algebra Appl. 651 (2022), 116-130. https://doi.org/10.1016/j.laa.2022.06.020
  14. D. Li and R. Qin, The Aα-spectral radius of graphs with a prescribed number of edges for ${\frac{1}{2}}$ ≤ α ≤ 1, Linear Algebra Appl. 628 (2021), 29-41. https://doi.org/10.1016/j.laa.2021.06.015
  15. S. Li and W. Sun, Some spectral inequalities for connected bipartite graphs with maximum Aα-index, Discrete Appl. Math. 287 (2020), 97-109. https://doi.org/10.1016/j.dam.2020.08.004
  16. S. Li and W. Sun, An arithmetic criterion for graphs being determined by their generalized Aα-spectra, Discrete Math. 344 (2021), no. 8, Paper No. 112469, 16 pp. https://doi.org/10.1016/j.disc.2021.112469
  17. S. Li and W. Sun, Some bounds on the Aα-index of connected graphs with fixed order and size, Linear Multilinear Algebra 70 (2022), no. 20, 5859-5878. https://doi.org/10.1080/03081087.2021.1932710
  18. S. Li, B. Tam, Y. Yu, and Q. Zhao, Connected graphs of fixed order and size with maximal Aα-index: the one-dominating-vertex case, Linear Algebra Appl. 662 (2023), 110-135. https://doi.org/10.1016/j.laa.2023.01.001
  19. S. Li and J. Wang, On the generalized Aα-spectral characterizations of almost α-ontrollable graphs, Discrete Math. 345 (2022), no. 8, Paper No. 112913, 17 pp. https://doi.org/10.1016/j.disc.2022.112913
  20. S. Li and W. Wei, The multiplicity of an Aα-eigenvalue: a unified approach for mixed graphs and complex unit gain graphs, Discrete Math. 343 (2020), no. 8, 111916, 19 pp. https://doi.org/10.1016/j.disc.2020.111916
  21. H. Lin, X. Huang, and J. Xue, A note on the Aα-spectral radius of graphs, Linear Algebra Appl. 557 (2018), 430-437. https://doi.org/10.1016/j.laa.2018.08.008
  22. H. Lin, X. G. Liu, and J. Xue, Graphs determined by their Aα-spectra, Discrete Math. 342 (2019), no. 2, 441-450. https://doi.org/10.1016/j.disc.2018.10.006
  23. 23] S. Liu, K. C. Das, and J. Shu, On the eigenvalues of Aα-matrix of graphs, Discrete Math. 343 (2020), no. 8, 111917, 12 pp. https://doi.org/10.1016/j.disc.2020.111917
  24. L. Lovasz and M. D. Plummer, Matching Theory, corrected reprint of the 1986 original, AMS Chelsea Publ., Providence, RI, 2009. https://doi.org/10.1090/chel/367
  25. V. Nikiforov, Merging the A- and Q-spectral theories, Appl. Anal. Discrete Math. 11 (2017), no. 1, 81-107. https://doi.org/10.2298/AADM1701081N
  26. V. Nikiforov and O. L. Rojo, On the α-index of graphs with pendent paths, Linear Algebra Appl. 550 (2018), 87-104. https://doi.org/10.1016/j.laa.2018.03.036
  27. S. Pirzada, Two upper bounds on the Aα-spectral radius of a connected graph, Commun. Comb. Optim. 7 (2022), no. 1, 53-57. https://doi.org/10.22049/CCO.2021.27061.1187
  28. P. Rowlinson, On the maximal index of graphs with a prescribed number of edges, Linear Algebra Appl. 110 (1988), 43-53. https://doi.org/10.1016/0024-3795(83)90131-3
  29. R. P. Stanley, A bound on the spectral radius of graphs with e edges, Linear Algebra Appl. 87 (1987), 267-269. https://doi.org/10.1016/0024-3795(87)90172-8
  30. L. Wang, X. Fang, X. Geng, and F. Tian, On the multiplicity of an arbitrary Aα-eigenvalue of a connected graph, Linear Algebra Appl. 589 (2020), 28-38. https://doi.org/10.1016/j.laa.2019.12.021
  31. C. X. Wang and T. She, A bound for the Aα-spectral radius of a connected graph after vertex deletion, RAIRO Oper. Res. 56 (2022), no. 5, 3635-3642. https://doi.org/10.1051/ro/2022176
  32. J. Wang, J. Wang, X. Liu, and F. Belardo, Graphs whose Aα-spectral radius does not exceed 2, Discuss. Math. Graph Theory 40 (2020), no. 2, 677-690. https://doi.org/10.7151/dmgt.2288
  33. W. Xi, W. So, and L. Wang, On the Aα spectral radius of digraphs with given parameters, Linear Multilinear Algebra 70 (2022), no. 12, 2248-2263. https://doi.org/10.1080/03081087.2020.1793879
  34. W. Xi and L. Wang, The Aα spectral radius and maximum outdegree of irregular digraphs, Discrete Optim. 38 (2020), 100592, 13 pp. https://doi.org/10.1016/j.disopt.2020.100592
  35. F. Xu, D. Wong, and F. Tian, On the multiplicity of α as an eigenvalue of the Aα matrix of a graph in terms of the number of pendant vertices, Linear Algebra Appl. 594 (2020), 193-204. https://doi.org/10.1016/j.laa.2020.02.025
  36. J. Xue, H. Lin, S. Liu, and J. Shu, On the Aα-spectral radius of a graph, Linear Algebra Appl. 550 (2018), 105-120. https://doi.org/10.1016/j.laa.2018.03.038
  37. J. Xue, R. Liu, G. Yu, and J. Shu, The multiplicity of Aα-eigenvalues of graphs, Electron. J. Linear Algebra 36 (2020), 645-657. https://doi.org/10.13001/ela.2020.4999
  38. G. Yu, On the maximal signless Laplacian spectral radius of graphs with given matching number, Proc. Japan Acad. Ser. A Math. Sci. 84 (2008), no. 9, 163-166. https://doi.org/10.3792/pjaa.84.163
  39. M. Zhai, J. Xue, and R. Liu, An extremal problem on Q-spectral radii of graphs with given size and matching number, Linear Multilinear Algebra 70 (2022), no. 20, 5334-5345. https://doi.org/10.1080/03081087.2021.1915231
  40. P. Zhang and X. D. Zhang, Lower bounds for the Aα-spectral radius of uniform hypergraphs, Linear Algebra Appl. 631 (2021), 308-327. https://doi.org/10.1016/j.laa.2021.08.021