Abstract
Let G be a graph with adjacency matrix A(G) and diagonal matrix of degrees Deg(G). For every real number \(\alpha \in [0,1],\) let \(A_\alpha (G)= \alpha {Deg}(G)+(1-\alpha )A(G).\) In this work, we consider the problem of ordering trees according to the spectral radius of the \(A_\alpha \)-matrices, the \(\alpha \)-index, determining those of order n that attain from the third to sixth largest values for this parameter. In particular, these results generalize the already known ordering when the spectral radius of the adjacency or Laplacian matrices are considered, instead of the \(\alpha \)-index. We also present several results concerning the \(\alpha \)-index of trees when the diameter or maximum degree are fixed parameters.
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Files used to support Conjecture 29 are available from the authors upon request.
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Funding
The research of the first, second and third author was partially supported by National Council for Scientific and Technological Development (CNPq) with CNPq Grant 141298/2016-2; CNPq Grant 403963/2021-4; CNPq Grant 313335/2020-6, respectively. The research of the second author was partially supported by FAPERJ with Grant E-20/2022-284573.
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Communicated by Leonardo de Lima.
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de Souza Maceira Belay, D., da Silva, C.M. & de Freitas, M.A.A. Ordering trees by \(\alpha \)-index. Comp. Appl. Math. 43, 20 (2024). https://doi.org/10.1007/s40314-023-02536-y
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DOI: https://doi.org/10.1007/s40314-023-02536-y