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

TL;DR

This paper generalizes the Graham-Pollak theorem by demonstrating that the hyperdeterminant of the Steiner distance hypermatrix is nonzero for even $k$, extending known results from the case $k=2$ to all even $k$.

Contribution

It extends the Graham-Pollak theorem to Steiner distances of even order, showing the hyperdeterminant's non-vanishing property for even $k$, and conjectures a dependence solely on $k$ and $n$.

Findings

Hyperdeterminant of Steiner distance hypermatrix is nonzero for even $k$.

The result generalizes the classical Graham-Pollak theorem.

Conjecture that the hyperdeterminant value depends only on $k$ and $n$.

Abstract

Graham and Pollak showed in 1971 that the determinant of a tree's distance matrix depends only on its number of vertices, and, in particular, it is always nonzero. The Steiner distance of a collection of $k$ vertices in a graph is the fewest number of edges in any connected subgraph containing those vertices; for $k=2$, this reduces to the ordinary definition of graphical distance. Here, we show that the hyperdeterminant of the $k$-th order Steiner distance hypermatrix is always nonzero if $k$ is even, extending their result beyond $k=2$. Previously, the authors showed that the $k$-Steiner distance hyperdeterminant is always zero for $k$ odd, so together this provides a generalization to all $k$. We conjecture that not just the vanishing, but the value itself, of the $k$-Steiner distance hyperdeterminant of an $n$-vertex tree depends only on $k$ and $n$.