  In this paper, we consider vertex-weighted digraphs with weights $(f,g)$ for
non-vanishing real functions $f$ and $g$ on the set of vertices. We study the
weighted analytic torsions. We calculate the weighted analytic torsions by
examples and prove that if the weight is of the diagonal form $(f,f)$, then the
weighted analytic torsion equals to the usual (un-weighted) analytic torsion.
