Abstract matrix-tree theorem
arXiv:1612.03873
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.
Version 2: added an application to the topology of 3-manifolds. A section about undirected graphs is completely rewritten. Many errors corrected. Version 3: an important reference is added