Weighted Analytic Torsions for Weighted Digraphs.

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.