The tau constant and the discrete Laplacian matrix of a metrized graph

Metrized graphs are finite graphs equipped with a distance function on their edges. For a metrized graph Γ, the tau constant τ(Γ) is an invariant which plays important roles in both harmonic analysis on metrized graphs and arithmetic of curves. T. Chinburg and R. Rumely [CR] introduced a canonical measure μcan of total mass 1 on a metrized graph Γ. The diagonal values of the Arakelov-Green’s function gμcan(x, x) associated to μcan are constant on Γ. M. Baker and Rumely called this constant “the tau constant” of a metrized graph Γ, and denoted it by τ(Γ). They [BR, Conjecture 14.5] posed a conjecture concerning the existence of a universal lower bound for τ(Γ). We call it Baker and Rumely’s lower bound conjecture. Baker and Rumely [BR] introduced a measure valued Laplacian operator ∆ which extends Laplacian operators studied earlier in [CR] and [Zh1]. This Laplacian operator combines the “discrete” Laplacian on a finite graph and the “continuous” Laplacian −f (x)dx on R. In terms of spectral theory, the tau constant τ(Γ) is the trace of the inverse operator of ∆ with respect to μcan when Γ has total length 1. The results in [Zh2] and the author’s thesis [C1, Chapter 4] indicate that the tau constant has important applications in arithmetic of curves such as its connection to the Effective Bogomolov Conjecture over function fields. In the article [C2], various formulas for τ(Γ) are given, and Baker and Rumely’s lower bound conjecture is verified for a number of large families of graphs. It is shown in the article [C3] that this conjecture holds for metrized graphs with edge connectivity more than 4; and proving it for cubic graphs is sufficient to show that it holds for all graphs. Verifying the Baker and Rumely’s lower bound conjecture in the remaining cases or showing a counter example to this conjecture, and finding metrized graphs with minimal tau constants are interesting and subtle problems. However, except for some special cases, computing the tau constant for metrized graphs with large number of vertices is not an easy task. In this paper, we will give a formula for the tau constant of Γ in terms of the discrete Laplacian matrix L of Γ and its pseudo inverse L. In particular, this formula leads to rapid computation of τ(Γ) by using computer softwares.