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Determinants of Steiner Distance Hypermatrices

Joshua Cooper, Zhibin Du

TL;DR

The paper extends the determinant theory from trees to order-$k$ Steiner distance hypermatrices by establishing a Graham–Lovász-style near-diagonalization that generalizes to $k>2$. It proves that the hyperdeterminant of the order-$k$ Steiner distance hypermatrix depends only on the tree size $n$ and the order $k$, and shows that the associated multilinear form is conditionally negative definite on the orthogonal complement of the all-ones vector for even $k$. A zeta/Möbius-based basis yields a near-diagonal representation independent of the specific tree, implying the determinant depends only on $n$ and $k$. The work also conjectures a fixed-sign behavior of the determinant and discusses partial results and open questions on positivity and eigenstructure in the even-$k$ regime, providing new tools and proofs for prior results in this tensor setting.

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-Lovász, 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.

Determinants of Steiner Distance Hypermatrices

TL;DR

The paper extends the determinant theory from trees to order- Steiner distance hypermatrices by establishing a Graham–Lovász-style near-diagonalization that generalizes to . It proves that the hyperdeterminant of the order- Steiner distance hypermatrix depends only on the tree size and the order , and shows that the associated multilinear form is conditionally negative definite on the orthogonal complement of the all-ones vector for even . A zeta/Möbius-based basis yields a near-diagonal representation independent of the specific tree, implying the determinant depends only on and . The work also conjectures a fixed-sign behavior of the determinant and discusses partial results and open questions on positivity and eigenstructure in the even- regime, providing new tools and proofs for prior results in this tensor setting.

Abstract

Generalizing work from the 1970s on the determinants of distance hypermatrices of trees, we consider the hyperdeterminants of order- Steiner distance hypermatrices of trees on vertices. We show that they can be nearly diagonalized as -forms, generalizing a result of Graham-Lovász, 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 and -- as Graham-Pollak showed for . We conclude with some open questions.
Paper Structure (4 sections, 14 theorems, 43 equations, 1 figure)

This paper contains 4 sections, 14 theorems, 43 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Action on $\mathbb{J}^\perp$
  3. Near-diagonalization
  4. Conclusion

Key Result

Theorem 2.1

For a tree $T$ on $n \geq 2$ vertices and a vector $\mathbf{c} \in \mathbb{R}^n$, write $\alpha_e = \mathbf{c}^T \mathbf{a}_e$ and $C = \mathbf{c}^T \mathbb{J}$. Then

Figures (1)

  • Figure 1: Depiction of the nearly diagonalized order-$4$ Steiner distance hypermatrix of any tree on $5$ vertices. Indices of the hypermatrix are represented by small and large numbers at the top and left of the diagram; gray represents a $0$ entry; nonzero entries are shown with their value ($-2$ or $\pm 1$).

Theorems & Definitions (29)

  • Theorem 2.1
  • proof
  • Corollary 2.2
  • proof
  • Corollary 2.3
  • proof
  • Definition 1
  • Theorem 2.4
  • proof
  • Theorem 2.5: Th11
  • ...and 19 more