Harmonic vectors and matrix tree theorems

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.

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Published 20 August 2014