A Generalization of the Graham-Pollak Tree Theorem to Steiner Distance

Joshua Cooper; Gabrielle Tauscheck

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arXiv · 2306.00243

A Generalization of the Graham-Pollak Tree Theorem to Steiner Distance

Joshua Cooper·Gabrielle Tauscheck

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Abstract

Graham and Pollak showed that the determinant of the distance matrix of a tree $T$ depends only on the number of vertices of $T$. Graphical distance, a function of pairs of vertices, can be generalized to ``Steiner distance'' of sets $S$ of vertices of arbitrary size, by defining it to be the fewest edges in any connected subgraph containing all of $S$. Here, we show that the same is true for trees' {\em Steiner distance hypermatrix} of all odd orders, whereas the theorem of Graham-Pollak concerns order $2$. We conjecture that the statement holds for all even orders as well.

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Joshua Cooper, Gabrielle Tauscheck. 2023-05-31. A Generalization of the Graham-Pollak Tree Theorem to Steiner Distance. https://arxiv.org/abs/2306.00243

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