Spectral Monotonicity under Leaf Attachment and Limiting Behavior in Discrete Einstein Trees
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| Format: | Preprint |
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2026
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| _version_ | 1866911708201615360 |
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| author | Bai, Shuliang Cheng, Haoxuan Hua, Bobo |
| author_facet | Bai, Shuliang Cheng, Haoxuan Hua, Bobo |
| contents | Let $R_T$ be the Ricci matrix of a finite tree $T$ introduced in \cite{BaiChengHua2026}, the largest eigenvalue $λ_{\max}(R_T)$ determines the sign of a discrete Einstein metric curvature on the tree. This paper investigates the asymptotic behavior of the sequence $λ_k = λ_{\max}(R_{T_k})$ obtained by repeatedly adding pendant edges at a fixed vertex. We prove that $λ_k$ converges to a limit $λ_\infty$ that depends only on the local branch data of $T$, and establish a first-order asymptotic expansion: \[ λ_k = λ_\infty + \fracα{d+k} + O\!\left(\frac{1}{(d+k)^2}\right), \] where $d$ is the degree of the original vertex, and the coefficient $α$ is given by a spectral projection. As a corollary, when $α\neq 0$, $λ_k$ is eventually strictly monotonic (increasing or decreasing). This theory reveals the fine influence of local leaf addition on the global spectrum. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_23379 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Spectral Monotonicity under Leaf Attachment and Limiting Behavior in Discrete Einstein Trees Bai, Shuliang Cheng, Haoxuan Hua, Bobo Differential Geometry Spectral Theory 53C21 Let $R_T$ be the Ricci matrix of a finite tree $T$ introduced in \cite{BaiChengHua2026}, the largest eigenvalue $λ_{\max}(R_T)$ determines the sign of a discrete Einstein metric curvature on the tree. This paper investigates the asymptotic behavior of the sequence $λ_k = λ_{\max}(R_{T_k})$ obtained by repeatedly adding pendant edges at a fixed vertex. We prove that $λ_k$ converges to a limit $λ_\infty$ that depends only on the local branch data of $T$, and establish a first-order asymptotic expansion: \[ λ_k = λ_\infty + \fracα{d+k} + O\!\left(\frac{1}{(d+k)^2}\right), \] where $d$ is the degree of the original vertex, and the coefficient $α$ is given by a spectral projection. As a corollary, when $α\neq 0$, $λ_k$ is eventually strictly monotonic (increasing or decreasing). This theory reveals the fine influence of local leaf addition on the global spectrum. |
| title | Spectral Monotonicity under Leaf Attachment and Limiting Behavior in Discrete Einstein Trees |
| topic | Differential Geometry Spectral Theory 53C21 |
| url | https://arxiv.org/abs/2605.23379 |