标题： 关于Graham-Pollak树定理的Steiner距离推广 显示英文标题

标题： A Generalization of the Graham-Pollak Tree Theorem to Steiner Distance

摘要： 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 size, by defining it to be the fewest edges in any connected subgraph containing all of $S$. Here, we show that the same is true for trees' {\em 斯坦纳距离超矩阵} 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. 显示英文摘要

摘要： 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 size, by defining it to be the fewest edges in any connected subgraph containing all of $S$. Here, we show that the same is true for trees' {\em 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.