Analytic torsion for graphs

TL;DR

This paper introduces an analytic torsion concept for graphs, linking it to linear algebra and the Dirac operator, with formulas for specific graph types like contractible graphs and discrete spheres.

Contribution

It defines analytic torsion for graphs using linear algebra and relates it to the Dirac operator, providing explicit formulas for certain classes of graphs.

Findings

Analytic torsion is the super determinant of the graph's Dirac operator.

Formulas are derived for contractible graphs and discrete spheres.

Analytic torsion generalizes the Ray-Singer torsion to graph structures.

Abstract

Analytic torsion is a functional on graphs which only needs linear algebra to be defined. In the continuum it corresponds to the Ray-Singer analytic torsion. We have formulas for analytic torsion if the graph is contractible or if it is a discrete sphere. A key insight is that analytic torsion is the super determinant of the Dirac operator of the graph.