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2025-06-02 10:07:49

This https://arxiv.org/abs/2407.01227 has been replaced.
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Non-intersecting paths and the determinant of the distance matrix of a tree
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.