Non-intersecting paths and the determinant of the distance matrix of a tree

TL;DR

This paper provides a combinatorial proof of the Graham-Pollak Formula for the determinant of a tree's distance matrix, unifying and extending existing generalizations using involutions and the Lindström-Gessel-Viennot Lemma.

Contribution

It offers the first combinatorial proof of the formula and introduces a unified framework for generalizations and q-analogues.

Findings

First combinatorial proof of the Graham-Pollak Formula

Unified framework for generalizations and q-analogues

Facilitates derivation of natural simultaneous generalizations

Abstract

We present the first combinatorial proof of the Graham-Pollak Formula for the determinant of the distance matrix of a tree, via sign-reversing involutions and the Lindstr\"om-Gessel-Viennot Lemma. Our approach provides a cohesive and unified framework for the understanding of the existing generalizations and $q$-analogues of the Graham-Pollak Formula, and facilitates the derivation of a natural simultaneous generalizations for them.