Families of Metrized Graphs With Small Tau Constants

2014/05/27 by Zübeyir Çınkır, Cinkir, Zubeyir
Computer Science · Mathematics · #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1405.7005

openalex publication_date 2014/05/27 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

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 (1)/(108) which is our conjectural lower bound.

Related

• The tau constant of a metrized graph and its behavior under graph operations
• The tau constant and the edge connectivity of a metrized graph
• The tau constant and the discrete Laplacian matrix of a metrized graph
• The Lower Bound for Number of Hexagons in Strongly Regular Graphs with Parameters λ=1 and μ=2
• A construction for bipatite Turán numbers