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

TL;DR

This paper extends the Graham-Pollak Tree Theorem to Steiner distances, showing that the determinant of the Steiner distance hypermatrix for trees depends only on the number of vertices, and conjectures this for all orders.

Contribution

It generalizes the Graham-Pollak theorem to Steiner distance hypermatrices of odd orders and proposes a conjecture for even orders.

Findings

Determinant of Steiner distance hypermatrix depends only on number of vertices for odd orders.

Theorem applies to all odd orders, extending classical results.

Conjecture that the property holds for all even orders.

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.