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

: $WW(G)=\frac 1 2 \sum_(d(u,v)+d^2(u,v)),$

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''.

Hyper-Wiener index can be used for the representation of computer networks and enhancing lattice hardware security. Hyper-Wiener indices used to limit the structure of a particle into a
solitary number which signifies the sub-atomic stretching
and electronic structures.

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,.
: $\operatorname(G_n)=20+\fracn+\fracn^2+\fracn^3+\fracn^4+\fracn^5+21n^6$

References

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Graph invariants