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Abstract
Metrized graphs, which are in 11 correspondence with weighted graphs, were introducedby R. Rumely in order to study arithmetic properties of curves. Reduction graphs, thedual graphs associated to the special fibre curve, are examples of metrized graphs. Rumelyand 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 ofmetrized graphs in the proof of Bogomolovs conjecture in the case of bad reductions.The diagonal value of the Arakelov-Greens function gcan(x, y) on a metrized graph Γ is a constant for a certain canonical measure µcan studied by Rumely and Chinburg ona 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 anumber of ways to describe τ (Γ). In terms of spectral theory, it is the trace of the inverse ofthe 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.