Note on the spectra of Steiner distance hypermatrices

TL;DR

This paper investigates the properties of Steiner distance hypermatrices in trees, revealing conditions under which their hyperdeterminant vanishes and exploring spectral radius characteristics, extending classical graph distance results.

Contribution

It extends classical distance matrix results to Steiner distance hypermatrices, providing new conditions for hyperdeterminant vanishing and spectral radius calculations.

Findings

Hyperdeterminant of Steiner distance hypermatrix vanishes under specific conditions

Distance spectral radius for n=2 equals 2^{k-1}-1

Conditions for hyperdeterminant vanishing depend on n and k

Abstract

The Steiner distance of a set of vertices in a graph is the fewest number of edges in any connected subgraph containing those vertices. The order-$k$ Steiner distance hypermatrix of an $n$-vertex graph is the $n \times \cdots \times n$ ($k$ terms) array indexed by vertices, whose entries are the Steiner distances of their corresponding indices. In the case of $k=2$, this reduces to the classical distance matrix of a graph. Graham and Pollak showed in 1971 that the determinant of the distance matrix of a tree only depends on its number $n$ of vertices. Here, we show that the hyperdeterminant of the Steiner distance hypermatrix of a tree vanishes if and only if (a) $n \geq 3$ and $k$ is odd, (b) $n=1$, or (c) $n=2$ and $k \equiv 1 \pmod{6}$. Two proofs are presented of the $n=2$ case -- the other situations were handled previously -- and we use the argument further to show that the distance spectral radius for $n=2$ is equal to $2^{k-1}-1$. Some related open questions are also discussed.