Trees with Extremal Hyper-Wiener Index: Mathematical Basis and Chemical Applications

Abstract

Trees with minimal and maximal hyper-Wiener indices (WW) are determined: Among n-vertex trees, minimum and maximum WW is achieved for the star-graph (Sn) and the path-graph (Pn), respectively. Since WW(Sn) is a quadratic polynomial in n,, whereas WW(Pn) is a quartic polynomial in n, the hyper-Wiener indices of all n-vertex trees assume values from a relatively narrow interval. Consequently, the hyper-Wiener index must have a very low isomer-discriminating power. This conclusion is corroborated by finding large families of trees, all members of which have equal WW-values.