Sums of Distances between Vertices/Leaves in K-ary Trees

Document Type


Publication Date


Publication Title

Bulletin of the Institute of Combinatorics and its Applications


As one of the most important topological indices emerged from the study of quantitative structure-activity relationships and quantitative structure-property relationships, the Wiener index (sum of the distances between vertices) has received great attention from both mathematicians and chemists. Recent study of the size of the tree bisection and reconnection neighborhood of phylogenetic binary trees inspired a rather similar concept, the sum of the distances of leaves of a tree. The extremal structures that maximize or minimize these two indices were studied and characterized for various classes of trees. An informal conjecture/question about the existence of a functional relation between these two indices naturally follows. A counterexample can be constructed for general trees, i.e. two trees T 1 and T 2 with the same order while T 1 has larger Wiener index and T 2 has larger sum of distances between leaves. In this note we provide a small but rather interesting observation on the correlations between these two indices among k-ary trees.