The tau constant of a metrized graph and its behavior under graph operations

TL;DR

This paper studies the tau constant of metrized graphs, providing formulas for its calculation, analyzing its behavior under various graph operations, and proving a key lower bound conjecture for specific graph classes.

Contribution

It introduces new formulas for the tau constant, examines its changes under graph modifications, and verifies Baker and Rumely's lower bound conjecture for certain graph classes.

Findings

Formulas for tau constant under graph operations

Behavior of tau constant when edges are replaced

Proof of lower bound conjecture for specific graphs

Abstract

This paper concerns the tau constant, which is an important invariant of a metrized graph, and which has applications to arithmetic properties of curves. We give several formulas for the tau constant, and show how it changes under graph operations including deletion of an edge, contraction of an edge, and union of graphs along one or two points. We show how the tau constant changes when edges of a graph are replaced by arbitrary graphs. We prove Baker and Rumely's lower bound conjecture on the tau constant for several classes of metrized graphs.