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

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 ( S n ) and the path-graph ( P n ), respectively. Since WW ( S n ) is a quadratic polynomial in n,, whereas WW ( P n ) 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.