  We prove the Effective Bogomolov Conjecture, and so the Bogomolov Conjecture,
over a function field of characteristic 0 by proving Zhang's Conjecture about
certain invariants of metrized graphs. In the function field case, these
conjectures were previously known to be true only for curves of genus at most 4
and a few other special cases. We also either verify or improve the previous
results. We relate the invariants involved in Zhang's Conjecture to the tau
constant of metrized graphs. Then we use and extend our previous results on the
tau constant. By proving another Conjecture of Zhang, we obtain a new proof of
the slope inequality for Faltings heights on moduli space of curves.
