Operators of rank 1, discrete path integration and graph Laplacians

TL;DR

This paper develops a formula for the characteristic polynomial of operators formed from rank 1 operators, using discrete path integration, generalizing known formulas for graph Laplacians and their determinants.

Contribution

It introduces a novel discrete path integration approach to express characteristic polynomials of certain operators, extending classical graph Laplacian formulas.

Findings

Generalizes the Forman-Kenyon formula for graph Laplacians

Provides a new determinant formula involving triangulated surfaces

Connects operator theory with combinatorial surface summations

Abstract

We prove a formula for a characteristic polynomial of an operator expressed as a polynomial of rank 1 operators. The formula uses a discrete analog of path integration and implies a generalization of the Forman-Kenyon's formula [4,6] for a determinant of the graph Laplacian (which, in its turn, implies the famous matrix-tree theorem by Kirchhoff) as well as its level 2 analog, where the summation is performed over triangulated nodal surfaces with boundary.