PreprintCC BY 4.0arXiv
The number of Laplacian eigenvalues of trees less than one
Abstract
Let mT[0,1) denote the number of Laplacian eigenvalues of a tree T that are strictly less than $1$. Guo, Xue and Liu \cite{GXL} proved that every tree T of diameter d satisfies mT[0,1)≥ ⌈(d+1)/3⌉, and that this bound is sharp when d≡2pmod3. It is classical that every graph of diameter d has domination number at least ⌈(d+1)/3⌉. In this paper, for any diameter d, mT[0,1)=⌈(d+1)/3⌉ if and only if γ(T)=⌈(d+1)/3⌉; we characterize all trees satisfying this necessary and sufficient condition, and also prove that almost all trees satisfy mT[0,1)≥⌈(d+1)/3⌉+1.
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BibTeX
@misc{guo2026number,
title = {{The number of Laplacian eigenvalues of trees less than one}},
author = {Jiaxin Guo and Tao Hu and Quanyu Tang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06899},
eprint = {2610.06899},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06899},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The number of Laplacian eigenvalues of trees less than one
AU - Jiaxin Guo
AU - Tao Hu
AU - Quanyu Tang
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.06899
UR - https://arxiv.org/abs/2610.06899
AB - Let m_T[0,1) denote the number of Laplacian eigenvalues of a tree T that are strictly less than $1$. Guo, Xue and Liu \cite{GXL} proved that every tree T of diameter d satisfies m_T[0,1)≥ ⌈(d+1)/3⌉, and that this bound is sharp when d≡2pmod3. It is classical that every graph of diameter d has domination number at least ⌈(d+1)/3⌉. In this paper, for any diameter d, m_T[0,1)=⌈(d+1)/3⌉ if and only if γ(T)=⌈(d+1)/3⌉; we characterize all trees satisfying this necessary and sufficient condition, and also prove that almost all trees satisfy m_T[0,1)≥⌈(d+1)/3⌉+1.
ER -