Edge Subdivision and the Perron Eigenvalue of Tree Ricci Matrices
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| Format: | Preprint |
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2026
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| _version_ | 1866913172285292544 |
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| author | Bai, Shuliang Cheng, Haoxuan Hua, Bobo |
| author_facet | Bai, Shuliang Cheng, Haoxuan Hua, Bobo |
| contents | The Ricci matrix $R_T$ of a finite tree encodes its discrete Einstein metrics via the Perron eigenvector, with Lin-Lu-Yau's Ollivier Ricci curvature: $κ= -λ_{\max}(R_T)$. We show that edge subdivision, the natural operation of lengthening a tree, can decrease, preserve, or increase $λ_{\max}$. Compressing each branch into a scalar feedback function via the Schur complement reduces the spectral problem to a one-dimensional Chebyshev equation. We obtain an exact one-step trichotomy, a scalar transmission equation for arbitrary length, and the long-chain limit. Examples on double stars, including an asymmetric case where subdivision strictly increases $λ_{\max}$, illustrate the theory. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_30949 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Edge Subdivision and the Perron Eigenvalue of Tree Ricci Matrices Bai, Shuliang Cheng, Haoxuan Hua, Bobo Differential Geometry Spectral Theory 53C21 The Ricci matrix $R_T$ of a finite tree encodes its discrete Einstein metrics via the Perron eigenvector, with Lin-Lu-Yau's Ollivier Ricci curvature: $κ= -λ_{\max}(R_T)$. We show that edge subdivision, the natural operation of lengthening a tree, can decrease, preserve, or increase $λ_{\max}$. Compressing each branch into a scalar feedback function via the Schur complement reduces the spectral problem to a one-dimensional Chebyshev equation. We obtain an exact one-step trichotomy, a scalar transmission equation for arbitrary length, and the long-chain limit. Examples on double stars, including an asymmetric case where subdivision strictly increases $λ_{\max}$, illustrate the theory. |
| title | Edge Subdivision and the Perron Eigenvalue of Tree Ricci Matrices |
| topic | Differential Geometry Spectral Theory 53C21 |
| url | https://arxiv.org/abs/2605.30949 |