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arXiv · 2605.30949

Edge Subdivision and the Perron Eigenvalue of Tree Ricci Matrices

Shuliang Bai·Haoxuan Cheng·Bobo Hua
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Abstract

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: $\kappa = -\lambda_{\max}(R_T)$. We show that edge subdivision, the natural operation of lengthening a tree, can decrease, preserve, or increase $\lambda_{\max}$. 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 $\lambda_{\max}$, illustrate the theory.

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Shuliang Bai, Haoxuan Cheng, Bobo Hua. 2026-05-29. Edge Subdivision and the Perron Eigenvalue of Tree Ricci Matrices. https://arxiv.org/abs/2605.30949

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