  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 $n$.
