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Most Aα-eigenvalues of a tree are small
Abstract
We study the distribution of Aα-eigenvalues of a tree. We extend the works \cite{Jacobs2021} and \cite{SIN2020} by proving that the number of Aα-eigenvalues, for 0≤ α≤ 1/2, less than or equal to the average degree dα= α(2− 2/n) is, at least, ⌈ n/2 ⌉ for a tree with n vertices. Several counterexamples for 1/2 < α≤ 1 are exhibited. We also derive the same result for other well-known families such as the Deformed Laplacian and the matrix Bβ.
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BibTeX
@misc{barturen2026most,
title = {{Most A$_α$-eigenvalues of a tree are small}},
author = {Elvia P. Barturén and Elismar R. Oliveira},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36427},
eprint = {2609.36427},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36427},
}
RIS
TY - UNPB
TI - Most A_α-eigenvalues of a tree are small
AU - Elvia P. Barturén
AU - Elismar R. Oliveira
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.36427
UR - https://arxiv.org/abs/2609.36427
AB - We study the distribution of A_α-eigenvalues of a tree. We extend the works \cite{Jacobs2021} and \cite{SIN2020} by proving that the number of A_α-eigenvalues, for 0≤ α≤ 1/2, less than or equal to the average degree d_α= α(2− 2/n) is, at least, ⌈ n/2 ⌉ for a tree with n vertices. Several counterexamples for 1/2 < α≤ 1 are exhibited. We also derive the same result for other well-known families such as the Deformed Laplacian and the matrix B_β.
ER -