On Kirchhoff's theorems with coefficients in a line bundle

TL;DR

This paper extends Kirchhoff's classical theorems to include 'twisted' versions involving chains with coefficients in a flat unitary line bundle, broadening their applicability in network analysis.

Contribution

It introduces twisted versions of Kirchhoff's theorems using chains with coefficients in a flat unitary line bundle, a novel generalization.

Findings

Proved twisted Kirchhoff's network theorem

Established twisted Kirchhoff's matrix-tree theorem

Extended classical results to line bundle coefficients

Abstract

We prove `twisted' versions of Kirchhoff's network theorem and Kirchhoff's matrix-tree theorem on connected finite graphs. Twisting here refers to chains with coefficients in a flat unitary line bundle.