English

Spectral Monotonicity under Leaf Attachment and Limiting Behavior in Discrete Einstein Trees

Differential Geometry 2026-05-25 v1 Spectral Theory

Abstract

Let RTR_T be the Ricci matrix of a finite tree TT introduced in \cite{BaiChengHua2026}, the largest eigenvalue λmax(RT)\lambda_{\max}(R_T) determines the sign of a discrete Einstein metric curvature on the tree. This paper investigates the asymptotic behavior of the sequence λk=λmax(RTk)\lambda_k = \lambda_{\max}(R_{T_k}) obtained by repeatedly adding pendant edges at a fixed vertex. We prove that λk\lambda_k converges to a limit λ\lambda_\infty that depends only on the local branch data of TT, and establish a first-order asymptotic expansion: λk=λ+αd+k+O ⁣(1(d+k)2), \lambda_k = \lambda_\infty + \frac{\alpha}{d+k} + O\!\left(\frac{1}{(d+k)^2}\right), where dd is the degree of the original vertex, and the coefficient α\alpha is given by a spectral projection. As a corollary, when α0\alpha \neq 0, λk\lambda_k is eventually strictly monotonic (increasing or decreasing). This theory reveals the fine influence of local leaf addition on the global spectrum.

Keywords

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

R2 v1 2026-07-22T07:27:51.670Z