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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.20862 |
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| _version_ | 1866910256671490048 |
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| author | Cheng, Haoxuan |
| author_facet | Cheng, Haoxuan |
| contents | For a weighted tree, the Lin--Lu--Yau Ricci curvature admits an explicit formula in terms of the edge weights. Consequently, the constant-curvature equation is equivalent to an eigenvalue problem for an edge-indexed Ricci matrix $R_T$. Building on the spectral characterization of discrete Einstein metrics on trees, we classify all finite trees whose discrete Einstein metric has positive curvature, equivalently all trees satisfying $λ_{\max}(R_T)<0$. For caterpillars with spine order $m\ge 12$, this occurs precisely for the endpoint families $T_m(a,0,\ldots,0,b)$ with $1\le a,b\le 3$ and $(a,b)\ne(3,3)$. The remaining cases $3\le m\le 11$ are settled by an exact finite verification using rational characteristic polynomials and Sturm root counts. We also determine the zero level set $λ_{\max}(R_T)=0$: among caterpillars, it consists of the stable family $(3,0,\ldots,0,3)$ together with nine exceptional short-spine caterpillars, while $S_3^2$ is the unique non-caterpillar zero example. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_20862 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Positive-Curvature Discrete Einstein Metrics on Trees Cheng, Haoxuan Differential Geometry Combinatorics 53C21 For a weighted tree, the Lin--Lu--Yau Ricci curvature admits an explicit formula in terms of the edge weights. Consequently, the constant-curvature equation is equivalent to an eigenvalue problem for an edge-indexed Ricci matrix $R_T$. Building on the spectral characterization of discrete Einstein metrics on trees, we classify all finite trees whose discrete Einstein metric has positive curvature, equivalently all trees satisfying $λ_{\max}(R_T)<0$. For caterpillars with spine order $m\ge 12$, this occurs precisely for the endpoint families $T_m(a,0,\ldots,0,b)$ with $1\le a,b\le 3$ and $(a,b)\ne(3,3)$. The remaining cases $3\le m\le 11$ are settled by an exact finite verification using rational characteristic polynomials and Sturm root counts. We also determine the zero level set $λ_{\max}(R_T)=0$: among caterpillars, it consists of the stable family $(3,0,\ldots,0,3)$ together with nine exceptional short-spine caterpillars, while $S_3^2$ is the unique non-caterpillar zero example. |
| title | Positive-Curvature Discrete Einstein Metrics on Trees |
| topic | Differential Geometry Combinatorics 53C21 |
| url | https://arxiv.org/abs/2605.20862 |