The spectral radius of Steiner distance hypermatrices of graphs

Abstract

Let G be an n-vertex connected graph with vertex set $V\left(G\right)$. Given a collection of k vertices (not necessarily distinct, which can be regarded as a tuple), say $S\in V{\left(G\right)}^k$, the Steiner distance $d_G\left(S\right)$ is defined as the fewest number of edges in any connected subgraph of G containing all the vertices in S. The Steiner distance would be reduced to the classical distance of two vertices in the case of $k=2$. Accordingly, one can generalize the distance matrix (with $k=2$) to the order-k hypermatrix, called Steiner distance hypermatrix, in which each entry is the Steiner distance of an n-dimensional array indexed by k vertices (not necessarily distinct). Very recently, Cooper and Tauscheck extended the classical Graham-Pollak theorem from the determinant of distance matrices of trees to the hyperdeterminant of Steiner distance hypermatrices of trees. In this paper, we consider the spectral radius of Steiner distance hypermatrices of general graphs, some extremal results are obtained.