## The tau constant of metrized graphs

##### Abstract

Metrized graphs, which are in 1−1 correspondence with weighted graphs, were introduced
by R. Rumely in order to study arithmetic properties of curves. “Reduction graphs”, the
dual graphs associated to the special fibre curve, are examples of metrized graphs. Rumely
and T. Chinburg used metrized graphs in their work on the “capacity pairing”. Later, S.
Zhang, in his work on “admissible pairing on curves”, demonstrated the important role of
metrized graphs in the proof of Bogomolov’s conjecture in the case of bad reductions.
The diagonal value of the Arakelov-Green’s function gµcan(x, y) on a metrized graph Γ
is a constant for a certain “canonical measure” µcan studied by Rumely and Chinburg on
a metrized graph Γ. This constant is called the “tau constant”, and was denoted τ (Γ), by
M.Baker and Rumely, who studied it in their Summer 2003 REU at UGA. There are a
number of ways to describe τ (Γ). In terms of spectral theory, it is the trace of the inverse of
the Laplacian on Γ with respect to µcan. Also, it is closely related to the resistance and the
voltage functions on Γ, when Γ is considered as an electrical network.
Our main focus in this thesis is to show the existence of a universal positive lower bound
to τ (Γ) for any Γ with length(Γ)=1. This is not completely achieved, but we prove it in
important cases and we develop a systematic theory of the tau constant. We give new interpretations of the canonical measure µcan in terms of the voltage functions on Γ. These enable
us to obtain new formulas for τ (Γ). We show how τ (Γ) changes under various graph operations including doubling edges, deleting and contracting edges, and taking unions of graphs
along one or two points. We establish some identities which we call the “deletion”, “contraction”, and “deletion-contraction” identities. We show that τ (Γ) ≥ length(Γ)·
(1− 4
λ
)
2
12 , if the
edge connectivity λ satisfies λ > 4. We show how τ (Γ) is related to the discrete Laplacian. We
present Maple codes and effective Matlab codes for computing τ (Γ). We also present several
families of graphs with equal edge lengths and having tau constants apparently approaching
to the value length(Γ)/108, which we conjecture to be the minimal possible value.
By using the properties of discrete Laplacian, we give new proofs of Foster’s identities
for electrical networks, and generalize them. We show, as an application, that there is an
explicit relation between τ (Γ) and the effective generalized Bogomolov’s conjectures over
global fields.