On the Hyperdeterminants of Steiner Distance Hypermatrices

Let $G$ be a graph on $n$ vertices. The Steiner distance of a collection of $k$ vertices in $G$ is the fewest number of edges in any connected subgraph containing those vertices. The order $k$ Steiner distance hypermatrix of $G$ is the $n$-dimensional array indexed by vertices, whose entries are the Steiner distances of their corresponding indices. In this paper, we confirm a conjecture on the Steiner distance hypermatrices proposed by Cooper and Du [Electron. J. Combin. 31(3):\#P3.4, 2024]. Furthermore, we also compute the hyperdeterminant of the order $k$ Steiner distance hypermatrix of $P_{3}$.