The Lure of the Distance Matrix of a Tree

Abstract: According to a classical result of Graham and Pollak, the determinant of the distance matrix of a tree depends only on the number of vertices and not on the tree. A formula for the inverse of the distance matrix was obtained by Graham and Lovasz. We discuss various results stemming from these facts. These include formulae for the mean first passage time of a random walk on a tree and the transfinite diameter of the metric space arising from a tree.