Wiener index, number of subtrees, and tree eccentric sequence

The eccentricity of a vertex $u$ in a connected graph $G$ is the distance between $u$ and a vertex farthest from it; the eccentric sequence of $G$ is the nondecreasing sequence of the eccentricities of $G$. In this paper, we determine the unique tree that minimises the Wiener index, i.e. the sum of distances between all unordered vertex pairs, among all trees with a given eccentric sequence. We show that the same tree maximises the number of subtrees among all trees with a given eccentric sequence, thus providing another example of negative correlation between the number of subtrees and the Wiener index of trees. Furthermore, we provide formulas for the corresponding extreme values of these two invariants in terms of the eccentric sequence. As a corollary to our results, we determine the unique tree that minimises the edge Wiener index, the vertex-edge Wiener index, the Schulz index (or degree distance), and the Gutman index among all trees with a given eccentric sequence.