Note on the spectra of Steiner distance hypermatrices
J Cooper, Z Du - arXiv preprint arXiv:2403.02287, 2024 - arxiv.org
J Cooper, Z Du
arXiv preprint arXiv:2403.02287, 2024•arxiv.orgThe 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$, this reduces to the classical distance matrix of a graph. Graham and Pollak
showed in 1971 that the determinant of the distance matrix of a tree only depends on its …
connected subgraph containing those vertices. The order-$ 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 …
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- Steiner distance hypermatrix of an -vertex graph is the ( terms) array indexed by vertices, whose entries are the Steiner distances of their corresponding indices. In the case of , 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 of vertices. Here, we show that the hyperdeterminant of the Steiner distance hypermatrix of a tree vanishes if and only if (a) and is odd, (b) , or (c) and . Two proofs are presented of the case -- the other situations were handled previously -- and we use the argument further to show that the distance spectral radius for is equal to . Some related open questions are also discussed.
arxiv.org