关键词:  Chain complex, Hodge-Laplace operator, Analytic torsion, Homology;
digraph, Weighted simplicial complex

Abstract:  In 2020, Alexander Grigor’yan, Yong Lin and Shing-Tung Yau [6] introduced
the Reidemeister torsion and the analytic torsion for digraphs by means of the path
complex and the path homology theory. Based on the analytic torsion for digraphs
introduced in [6], we consider the notion of weighted analytic torsion for vertex-weighted
digraphs. For any non-vanishing real functions f and g on the vertex set, we consider the
vertex-weighted digraphs with the weights ( f,g ). We calculate the ( f,g )-weighted analytic
torsion by examples and prove that the ( f,g )-weighted analytic torsion only depend on
the ratio f/g . In particular, if the weight is of the diagonal form ( f,f ), then the weighted
analytic torsion equals to the usual (un-weighted) torsion.

Key words:  Chain complex, Hodge-Laplace operator, Analytic torsion, Homology;
digraph, Weighted simplicial complex