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

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

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

Combinatorics 2025-06-02 v2

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

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.

Keywords

distance matrix tree non-intersecting paths Lindstr"om-Gessel-Viennot lemma combinatorial proof

Cite

@article{arxiv.2407.01227,
  title  = {Non-intersecting paths and the determinant of the distance matrix of a tree},
  author = {Emmanuel Briand and Luis Esquivias-Quintero and Álvaro Gutiérrez and Adrián Lillo and Mercedes Rosas},
  journal= {arXiv preprint arXiv:2407.01227},
  year   = {2025}
}

Comments

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

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

Abstract

Keywords

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