The distances between internal vertices and leaves of a tree

Abstract

Distance-based graph invariants have been of great interest and extensively studied. The classic Wiener index was proposed in biochemistry and defined to be the sum of distances between all pairs of vertices. The sum of distances between all pairs of leaves, named the Gamma index and the terminal Wiener index respectively, was motivated from both biochemistry and phylogenetic reconstruction. The studies of such distance-based graph invariants include, for instance, the “middle part” of a tree, the extremal structures with given constraints, the extremal values of ratios of the distance function at the “middle part” and leaves. In particular, when considering the extremal structures, correlations between the Wiener index and other graph invariants have been discovered and general methods have been developed. Other related graph invariants include the number of subtrees or leaf containing subtrees (corresponding to the “acceptable residue configurations”), subforests, root-containing subtrees (in relation to the antichains in a Hasse diagram with the structure of a rooted tree), to name a few.As has been repeatedly mentioned in earlier works, it is of both mathematical interests and practical importance to generalize the current studies to other distance-based graph invariants. The sum of distances between internal vertices has been considered and many similar results have been obtained.On the other hand, a natural similar concept is the sum of distances between internal vertices and leaves. It has been proposed in different literatures. In this paper we conduct a relatively comprehensive study of this concept, providing results analogous to those of the aforementioned studies. We start with identifying the “middle part” of a tree with respect to the total distance from leaves. Then the extremal ratios with respect to the distance from leaves are examined and the structures achieving the extremal ratios are characterized. Lastly, we provide extremal trees under different constraints that maximize or minimize the sum of all such distances.