Higher determinants and the matrix-tree theorem
Combinatorics
2016-12-14 v2
Abstract
The classical matrix-tree theorem was discovered by G.~Kirchhoff in 1847. It relates the principal minor of the Laplace (nxn)-matrix to a particular sum of monomials indexed by the set of trees with n vertices. The aim of this paper is to present a generalization of the (nonsymmetric) matrix-tree theorem containing no trees and essentially no matrices. Instead of trees we consider acyclic directed graphs with a prescribed set of sinks, and instead of determinant, a polynomial invariant of the matrix determined by directed graph such that any two vertices of the same connected component are mutually reacheable.
Keywords
Cite
@article{arxiv.1508.02245,
title = {Higher determinants and the matrix-tree theorem},
author = {Yurii Burman},
journal= {arXiv preprint arXiv:1508.02245},
year = {2016}
}
Comments
Most of the material, with better proofs, was superseded by the later preprint http://arxiv.org/abs/1612.03873 ; take it instead