Contents Online

# Journal of Combinatorics

## Volume 5 (2014)

### Number 2

### Harmonic vectors and matrix tree theorems

Pages: 195 – 202

DOI: http://dx.doi.org/10.4310/JOC.2014.v5.n2.a3

#### Author

#### Abstract

This 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 recent 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 weighted 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.

#### Keywords

market mechanisms, discrete Laplacian, matrix tree theorem, all minors

#### 2010 Mathematics Subject Classification

Primary 05A15. Secondary 05C22, 05C30, 91B24, 91B26.