Harmonic vectors and matrix tree theorems

Abstract

The paper describes an explicit combinatorial formula for a harmonic vector for the Laplacian of a directed graph with arbitrary edge weights. This result was motivated by questions from mathematical economics, and the formula plays a crucial role in forthcoming joint work of the author on the emergence of prices and money in an exchange economy. It turns out that the formula is closely related to well-studied problems in graph theory, in particular to the so-called matrix tree theorem due to W. Tutte and independently to R. Bott and J. Mayberry. As a further application of our considerations, we obtain a short new proof of both the matrix tree theorem as well as its generalization due to S. Chaiken.