Operators of rank 1, discrete path integration and graph Laplacians
arXiv:1205.1123
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.
17 pages, 1 figure. After version 2 two references were added, introduction improved. Major changes (new results added, rewritten completely) after version 1