Principal minors of the distance matrix of a tree

TL;DR

This paper provides a combinatorial proof for a formula to compute principal minors of the distance matrix of a tree, extending previous work by Graham, Pollak, and others.

Contribution

It introduces the first combinatorial proof for the principal minors formula of a tree's distance matrix, generalizing earlier algebraic results.

Findings

Derived a combinatorial formula for principal minors of tree distance matrices

Extended previous algebraic results to combinatorial proofs

Generalized Graham and Pollak's work on distance matrices

Abstract

Let $T = ([n], E)$ be a tree and let $D = ( d(i,j) )_{i, j \le n}$ be the distance matrix of $T$. Let $S\subseteq [n]$. We give the first combinatorial proof for a formula to compute the principal minor of $D$ indexed by $S$, namely $\det D[S]$. This generalizes work of Graham and Pollak, as well as more recent works.