Non-intersecting Paths and the Determinant of the Distance Matrix of a Tree

Emmanuel Briand, Luis Esquivias-Quintero, Álvaro Gutiérrez, Adrián Lillo, Mercedes Rosas

Published 2024-07-01Version 1

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.