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

Joshua Cooper, Gabrielle Tauscheck

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 $n$.

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

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 vertices in a graph is the fewest number of edges in any connected subgraph containing those vertices; for , this reduces to the ordinary definition of graphical distance. Here, we show that the hyperdeterminant of the -th order Steiner distance hypermatrix is always nonzero if is even, extending their result beyond . Previously, the authors showed that the -Steiner distance hyperdeterminant is always zero for odd, so together this provides a generalization to all . We conjecture that not just the vanishing, but the value itself, of the -Steiner distance hyperdeterminant of an -vertex tree depends only on and .
Paper Structure (2 sections, 6 theorems, 18 equations)

This paper contains 2 sections, 6 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Main Results

Key Result

Theorem 1.1

The hyperdeterminant $\det(M)$ of the order-$k$, dimension-$n$ hypermatrix $M = (M_{i_1,\ldots,i_k})_{i_1,\ldots,i_k=1}^n$ is a monic irreducible polynomial which evaluates to zero iff there is a nonzero simultaneous solution to $\nabla f_M = \vec{0}$, where

Theorems & Definitions (16)

  • Theorem 1.1: GelKapZel08 Theorem 1.3
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • ...and 6 more