  Several invariants of polarized metrized graphs and their applications in
Arithmetic Geometry are studied recently. In this paper, we explicitly
calculated these admissible invariants for all curves of genus $3$. We find the
sharp lower bound for the invariants $\varphi$, $\lambda$ and $\epsilon$ for
all polarized metrized graphs of genus $3$. This improves the lower bound given
for Effective Bogomolov Conjecture for such curves.
