Operators of rank 1, discrete path integration and graph Laplacians

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.