The Distance Between the Adjacency Spectral Center and the Characteristic Set of a Tree
Abstract
Let 𝒮(T) be the adjacency spectral center of a tree T, and let 𝒞(T) be its characteristic set. We determine the largest possible separation d(T) := distT(𝒮(T), 𝒞(T)) among trees of every order n≥3. Writing Δn:=max|V(T)|=nd(T), we prove Δn=0 (3≤ n≤11), Δ12=1, and Δn=⌊(n−11)/2⌋ (n≥13). The argument rests on a simple opposition between two rooted-tree weights. An endpoint-rooted path minimizes adjacency spectral radius, but maximizes bottleneck Perron value. A one-sided replacement by a path therefore cannot decrease the distance between the two centers. Quantitatively, this gives the sharp estimate |V(T)|≥ 2d(T)+11 (d(T)≥2). A preliminary six-vertex barrier shows that disjoint center sets require at least twelve vertices, and the four-leaf broom is extremal for every n≥12.
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BibTeX
@misc{zhu2026distance,
title = {{The Distance Between the Adjacency Spectral Center and the Characteristic Set of a Tree}},
author = {Chaochao Zhu and Shipei Hu and Jingfu Huang and Qin Yue},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16943},
eprint = {2609.16943},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16943},
}
RIS
TY - UNPB
TI - The Distance Between the Adjacency Spectral Center and the Characteristic Set of a Tree
AU - Chaochao Zhu
AU - Shipei Hu
AU - Jingfu Huang
AU - Qin Yue
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16943
UR - https://arxiv.org/abs/2609.16943
AB - Let 𝒮(T) be the adjacency spectral center of a tree T, and let 𝒞(T) be its characteristic set. We determine the largest possible separation d(T) := dist_T(𝒮(T), 𝒞(T)) among trees of every order n≥3. Writing Δ_n:=max_|V(T)|=nd(T), we prove Δ_n=0 (3≤ n≤11), Δ_12=1, and Δ_n=⌊(n−11)/2⌋ (n≥13). The argument rests on a simple opposition between two rooted-tree weights. An endpoint-rooted path minimizes adjacency spectral radius, but maximizes bottleneck Perron value. A one-sided replacement by a path therefore cannot decrease the distance between the two centers. Quantitatively, this gives the sharp estimate |V(T)|≥ 2d(T)+11 (d(T)≥2). A preliminary six-vertex barrier shows that disjoint center sets require at least twelve vertices, and the four-leaf broom is extremal for every n≥12.
ER -