Wiener Indices of Balanced Binary Trees

Molecules and molecular compounds are often modeled by molecular graphs. One of the most widely known topological descriptor [6, 9] is the Wiener index named after chemist Harold Wiener [13]. The Wiener index of a graph G(V, E) is defined as W (G) = ∑ u,v∈V d(u, v), where d(u, v) is the distance between vertices u and v (minimum number of edges between u and v). A majority of the chemical applications of the Wiener index deal with chemical compounds that have acyclic organic molecules. The molecular graphs of these compounds are trees [7], see an example of a chemical compound in Fig. 1. Therefore most of the prior work on the Wiener indices deals with trees, relating the structure of various trees to their Wiener indices (asymptotic bounds on the Wiener indices of certain families of trees, expected Wiener indices of random trees etc.). For these reasons, we concentrate on the Wiener indices of trees as well (see Dobrynin et al. [3] for a recent survey). For trees with bounded degrees of vertices, Jelen and Triesch [10] found a family of trees such that W (T ) is minimized. Fischermann et al. [4] solved the same problem independently. They characterized the trees that minimize and maximize the Wiener index among all trees of a given size and the maximum vertex degree. Several papers address the question: What positive integer numbers can be Wiener indices of graphs of a certain type? The question is answered for general graphs and bipartite graphs [3]. The question is still open for trees.