Edge Subdivision and the Perron Eigenvalue of Tree Ricci Matrices

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

17 pages, 1 figure