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.
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Briand et al. (2024) studied this question.
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