Principal minors of tree distance matrices

Harry Richman, Farbod Shokrieh, Chenxi Wu

Published 2024-11-18Version 1

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. 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.