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arXiv · 2407.01227

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

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

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Emmanuel Briand, Luis Esquivias-Quintero, Álvaro Gutiérrez, Adrián Lillo, Mercedes Rosas. 2025-05-30. Non-intersecting paths and the determinant of the distance matrix of a tree. https://arxiv.org/abs/2407.01227

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