Principal minors of tree distance matrices

TL;DR

This paper provides a combinatorial formula for the principal minors of tree distance matrices, extending previous results and using potential theory and Symanzik polynomials.

Contribution

It generalizes Graham and Pollak's result and refines Graham and Lovász's work by expressing minors in terms of rooted spanning forests and potential theory.

Findings

Principal minors relate to counts of rooted spanning forests.

Extended formulas to trees with edge lengths.

Connections to Symanzik polynomial evaluations.

Abstract

We prove that the principal minors of the distance matrix of a tree satisfy a combinatorial expression involving counts of rooted spanning forests of the underlying tree. This generalizes a result of Graham and Pollak, and refines a result of Graham and Lov\'asz on the coefficients of the characteristic polynomial of the distance matrix. We also give such an expression for the case of trees with edge lengths. We use arguments motivated by potential theory on graphs. Our formulas can be expressed in terms of evaluations of Symanzik polynomials.