Hyper-Wiener index

The hyper-Wiener index is a generalization introduced by Milan Randić [1] of the concept of the Wiener index, introduced by Harry Wiener.

The hyper-Wiener index of a connected graph G is defined by where d(u,v) is the distance between vertex u and v. Hyper-Wiener index as topological index assigned to G = (V,E) is based on the distance function which is invariant under the action of the automorphism group of G. Hyper-Wiener index can be used for the representation of computer networks and enhancing lattice hardware security.

One-pentagonal carbon nanocone which is an infinite symmetric graph, consists of one pentagon as its core surrounded by layers of hexagons.

If there are n layers, then the graph of the molecules is denoted by Gn.

we have the following explicit formula for hyper-Wiener index of one-pentagonal carbon nanocone,[2]