Families of Metrized Graphs With Small Tau Constants

TL;DR

This paper constructs families of metrized graphs with small tau constants, providing numerical evidence and proving that certain graphs asymptotically approach a conjectured lower bound, supporting Baker and Rumely's conjecture.

Contribution

It introduces new families of metrized graphs with small tau constants and proves asymptotic behavior towards the conjectured lower bound.

Findings

Families of graphs with small tau constants constructed

Numerical computations support small tau constants

Hexagonal nets asymptotically approach the lower bound of 1/108

Abstract

Baker and Rumely's tau lower bound conjecture claims that if the tau constant of a metrized graph is divided by its total length, this ratio must be bounded below by a positive constant for all metrized graphs. We construct several families of metrized graphs having small tau constants. In addition to numerical computations, we prove that the tau constants of the metrized graphs in one of these families, the hexagonal nets around a torus, asymptotically approach to $\frac{1}{108}$ which is our conjectural lower bound.