Document Type
Article
Abstract
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-k Steiner distance hypermatrix of a graph is the n-dimensional 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 number n of vertices. Here, we show that the hyperdeterminant of the Steiner distance hypermatrix of a tree vanishes if and only if (a) n ≧ 3 and k is odd, (b) n = 1, or (c) n = 2 and k ≡ 1 (mod 6). Two proofs are presented of the n = 2 case – the other situations were handled previously – and we use the argument further to show that the distance spectral radius for n = 2 is equal to 2k−1 − 1. Some related open questions are also discussed.
Digital Object Identifier (DOI)
Publication Info
Published in Electronic Journal of Combinatorics, Volume 31, Issue 3, 2024.
APA Citation
Cooper, J., & Du, Z. (2024). Note on the Spectra of Steiner Distance Hypermatrices. The Electronic Journal of Combinatorics, 31(3).https://doi.org/10.37236/12850
Rights
©The authors. Released under the CC BY-ND licenseInternational 4.0.