Weighted Analytic Torsion for Weighted Digraphs

REN Shi-quan, WANG Chong

doi:10.13371/j.cnki.chin.q.j.m.2023.01.003

Chinese Quarterly Journal of Mathematics >

2023 , Vol. 38 >Issue 1: 30 - 49

DOI: https://doi.org/10.13371/j.cnki.chin.q.j.m.2023.01.003

Weighted Analytic Torsion for Weighted Digraphs

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1. School of Mathematics and Statistics, Henan University, Kaifeng 475004, China; 2. School of
Mathematics and Statistics, Cangzhou Normal University, Cangzhou 061000, China

REN Shi-quan (1987-), male, native of Jinan, Shandong, lecturer of Henan University, engages in topology; WANG Chong (1981-), female, native of Baoding, Hebei, associate professor of Cangzhou Normal University, engages in topology.

Received date: 2023-04-11

Online published: 2023-03-20

Supported by

REN Shi-quan is supported by China Postdoctoral Science Foundation (Grant No.
2022M721023); WANG Chong is supported by Science and Technology Project of Hebei Education Department
(Grant No. ZD2022168) and Project of Cangzhou Normal University (Grant No. XNJJLYB2021006).

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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

Cite this article

REN Shi-quan, WANG Chong . Weighted Analytic Torsion for Weighted Digraphs[J]. Chinese Quarterly Journal of Mathematics, 2023 , 38(1) : 30 -49 . DOI: 10.13371/j.cnki.chin.q.j.m.2023.01.003

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