Spectral Monotonicity under Leaf Attachment and Limiting Behavior in Discrete Einstein Trees
Differential Geometry
2026-05-25 v1 Spectral Theory
Abstract
Let be the Ricci matrix of a finite tree introduced in \cite{BaiChengHua2026}, the largest eigenvalue determines the sign of a discrete Einstein metric curvature on the tree. This paper investigates the asymptotic behavior of the sequence obtained by repeatedly adding pendant edges at a fixed vertex. We prove that converges to a limit that depends only on the local branch data of , and establish a first-order asymptotic expansion: where is the degree of the original vertex, and the coefficient is given by a spectral projection. As a corollary, when , is eventually strictly monotonic (increasing or decreasing). This theory reveals the fine influence of local leaf addition on the global spectrum.
Cite
@article{arxiv.2605.23379,
title = {Spectral Monotonicity under Leaf Attachment and Limiting Behavior in Discrete Einstein Trees},
author = {Shuliang Bai and Haoxuan Cheng and Bobo Hua},
journal= {arXiv preprint arXiv:2605.23379},
year = {2026}
}
Comments
15 pages, 2 figures