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.