The Tau Constant of a Metrized Graph and its Behavior under Graph Operations

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.

Families of Metrized Graphs with Small Tau Constants

“…We consider Equation (1) One can find more detailed information on τ (Γ) in articles [6], [7], [8] and [9]. Let Γ − e i be a connected graph for an edge e i ∈ E(Γ) of length L i .…”

“…Whenever we have a vertex p ∈ V (Γ) with υ(p) ≥ 3, considering points q ∈ Γ with υ(q) = 2 as a vertex or not does not change Γ, which is called the valence property of tau constant (see [7,Remark 2.10]). Therefore, in our search of small tau constants, we work with metrized graphs that we can keep V (Γ) minimal by considering only the points with valence at least 3 as vertices.…”

“…where the union is taken along the vertex p. This is called the additive property of the tau constant (see [7,Page 15]). As a consequence of this property, we can restrict our attention to the metrized graphs with vertex connectivity at least 2 to find smaller tau constants.…”

“…The tau constant τ (Γ) of a metrized graph Γ is an invariant which plays important roles in both harmonic analysis on metrized graphs [4] and arithmetic of curves [10]. Its properties are studied in the articles [7] and [9]. Moreover, its algorithmic computation is given in [8].…”

“…In this article, we first review some background material in §2. In §3, we gather various related results about tau constant (from [4], [3], [6], [7], [8], [9], [11]) to obtain necessary conditions to have small tau constants. Using the conditions we obtained in this article and interpreting our previous results, we develop a strategy to search small tau constants.…”