---
abstract: 'The classical matrix-tree theorem discovered by G.Kirchhoff in 1847 relates the principal minor of the $n \times n$ 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.'
address: 'National Research University Higher School of Economics, 119048, 6 Usacheva str., Moscow, Russia, and Independent University of Moscow, 119002, 11 B.Vlassievsky per., Moscow, Russia'
author:
- Yurii Burman
title: 'Abstract matrix-tree theorem'
---

Introduction and the main results
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Principal definitions {#SSec:Def}
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Denote by $\Gamma_{n,k}$ the set of all directed graphs with $n$ vertices numbered $1 {, \dots ,} n$ and $k$ edges numbered $1 {, \dots ,} k$. We will write $e =
[ab]$ if $e$ is an edge from vertex $a$ to vertex $b$; in particular, $[aa]$ means a loop attached to the vertex $a$. We will treat elements of $\Gamma_{n,k}$ as sequences of 