Determinants of Steiner Distance Hypermatrices

TL;DR

This paper extends the study of distance hypermatrices of trees to Steiner distance hypermatrices, revealing their hyperdeterminants' structure, generalizing classical results, and confirming they depend only on the parameters $k$ and $n$.

Contribution

It introduces the hyperdeterminants of Steiner distance hypermatrices, generalizes diagonalization results, and confirms their dependence solely on $k$ and $n$, resolving a conjecture.

Findings

Hyperdeterminants can be nearly diagonalized as $k$-forms.

Provides a tensor version of 'conditional negative definiteness'.

Confirms hyperdeterminants depend only on $k$ and $n$.

Abstract

Generalizing work from the 1970s on the determinants of distance hypermatrices of trees, we consider the hyperdeterminants of order-$k$ Steiner distance hypermatrices of trees on $n$ vertices. We show that they can be nearly diagonalized as $k$-forms, generalizing a result of Graham-Lov\'{a}sz, implying a tensor version of ``conditional negative definiteness'', providing new proofs of previous results of the authors and Tauscheck, and resolving the conjecture that these hyperdeterminants depend only on $k$ and $n$ -- as Graham-Pollak showed for $k=2$. We conclude with some open questions.