A generalization of the Graham-Pollak tree theorem to Steiner distance

Abstract

Graham and Pollak showed that the determinant of the distance matrix of a tree T depends only on the number of vertices of T. Graphical distance, a function of pairs of vertices, can be generalized to “Steiner distance” of sets S of vertices of arbitrary cardinality, by defining it to be the minimum number of edges in a connected subgraph containing all the vertices of S. Here, we show that the same is true for trees' Steiner distance hypermatrix of all odd orders, whereas the theorem of Graham-Pollak concerns order 2. We conjecture that the statement holds for all even orders as well.