The tau constant and the edge connectivity of a metrized graph

2009/01/12 by Zübeyir Çınkır, Zubeyir Cinkir, Cinkir, Zubeyir · 1 citation
Computer Science · Mathematics · #Advanced Graph Theory Research #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Limits and Structures in Graph Theory #math.CA #math.CO

paper · pdf · doi:10.48550/arxiv.0901.1481

Various typos are corrected. Some minor changes made to the content. However, all the results remain the same. 27 pages, 6 figues

openalex publication_date 2009/01/12 · arxiv created 2009/05/20 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The tau constant is an important invariant of a metrized graph, and it has applications in arithmetic properties of curves. We show how the tau constant of a metrized graph changes under successive edge contractions and deletions. We discover identities which we call "contraction", "deletion", and "contraction-deletion" identities on a metrized graph. By establishing a lower bound for the tau constant in terms of the edge connectivity, we prove that Baker and Rumely's lower bound conjecture on the tau constant holds for metrized graphs with edge connectivity 5 or more. We show that proving this conjecture for 3-regular graphs is enough to prove it for all graphs.

Cited by

• The tau constant of a metrized graph and its behavior under graph operations
• Zhang's Conjecture and the Effective Bogomolov Conjecture over function fields

Related

• The tau constant of a metrized graph and its behavior under graph operations
• Families of Metrized Graphs With Small Tau Constants
• The tau constant and the discrete Laplacian matrix of a metrized graph
• Construction Sequences and Certifying 3-Connectedness
• Strict inequalities for connective constants of transitive graphs