Higher determinants and the matrix-tree theorem

TL;DR

This paper generalizes the classical matrix-tree theorem by introducing a polynomial invariant for directed acyclic graphs with sinks, moving beyond trees and matrices to broader combinatorial structures.

Contribution

It presents a novel generalization of the matrix-tree theorem involving acyclic directed graphs and a new polynomial invariant, without relying on traditional matrices or trees.

Findings

Introduces a polynomial invariant for directed graphs.

Extends the matrix-tree theorem to acyclic directed graphs.

Provides a new combinatorial framework beyond classical matrices.

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.