A Classification of Positive-Curvature Discrete Einstein Metrics on Trees

TL;DR

This paper classifies all finite trees with positive discrete Einstein metrics based on spectral properties of an associated Ricci matrix, providing explicit characterizations for caterpillars and other trees.

Contribution

It offers a complete classification of finite trees with positive curvature discrete Einstein metrics, including explicit families and verification methods.

Findings

Caterpillars with spine order ≥ 12 have positive curvature for specific endpoint families.

Exact finite verification determines cases with spine order 3 to 11.

Zero curvature cases include a stable family and nine exceptional caterpillars.

Abstract

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 $\lambda_{\max}(R_T)<0$. 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 $\lambda_{\max}(R_T)=0$: 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.