Discrete Einstein metrics on trees

TL;DR

This paper proves the existence and uniqueness of discrete Einstein metrics on trees with positive curvature, revealing structural properties and monotonicity of edge weights.

Contribution

It introduces a novel approach using Perron-Frobenius theory to establish discrete Einstein metrics on trees and characterizes their geometric features.

Findings

Positive-curvature Einstein metric implies the tree is a caterpillar.

Edge weights decrease strictly away from the maximal edge.

Existence and uniqueness of these metrics are established.

Abstract

We establish the existence and uniqueness of discrete Einstein metrics on trees under Lin-Lu-Yau Ricci curvature using Perron-Frobenius theory. Notably, the existence of a positive-curvature Einstein metric implies the tree must be a caterpillar. Furthermore, these metrics exhibit radial monotonicity, with edge weights decreasing strictly away from the maximal edge.