Hyper-Wiener index

In chemical graph theory, the hyper-Wiener index or hyper-Wiener number is a topological index of a molecule, used in biochemistry. The hyper-Wiener index is a generalization introduced by Milan Randić of the concept of the Wiener index, introduced by Harry Wiener. The hyper-Wiener index of a connected graph G is defined by

$WW(G)={\frac {1}{2}}\sum _{u,v\in V(G)}(d(u,v)+d^{2}(u,v)),$

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.

Example

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 G_n. we have the following explicit formula for hyper-Wiener index of one-pentagonal carbon nanocone,

$\operatorname {WW} (G_{n})=20+{\frac {533}{4}}n+{\frac {8501}{24}}n^{2}+{\frac {5795}{12}}n^{3}+{\frac {8575}{24}}n^{4}+{\frac {409}{3}}n^{5}+21n^{6}$

pentagonal-carbon-nanocone