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

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 Lindström-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.

Changed the way in which networks are presented, in hopes of simplification and rigour. Technical details postponed to later sections. 37 pages, 15 figures