Abstract matrix-tree theorem

TL;DR

This paper generalizes Kirchhoff's classical matrix-tree theorem by defining a polynomial det_{n,k} that relates to sums over acyclic graphs with n vertices and k edges, extending the original theorem.

Contribution

It introduces a new polynomial det_{n,k} that generalizes the matrix-tree theorem to graphs with more edges, linking algebraic and combinatorial structures.

Findings

The polynomial det_{n,k} applied to the Laplace matrix sums over acyclic graphs.

Extension of the classical matrix-tree theorem to graphs with k edges.

Provides a new algebraic-combinatorial relationship for graph enumeration.

Abstract

The classical matrix-tree theorem discovered by G.Kirchhoff in 1847 relates the principal minor of the nxn Laplace matrix to a particular sum of monomials of matrix elements indexed by directed trees with n vertices and a single sink. In this paper we consider a generalization of this statement: for any k \ge n we define a degree k polynomial det_{n,k} of matrix elements and prove that this polynomial applied to the Laplace matrix gives a sum of monomials indexed by acyclic graphs with n vertices and k edges.