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 ^[1] of the concept of the Wiener index, introduced by Harry Wiener. The hyper-Wiener index of a connected graph G is defined by

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

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.

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,^[2]

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

pentagonal-carbon-nanocone

References

1. Randic, M. (1993), "Novel molecular descriptor for structure—property studies", Chemical Physics Letters, 211 (10): 478–483, Bibcode:1993CPL...211..478R, doi:10.1016/0009-2614(93)87094-J.
2. Darafsheh, M. R.; Khalifeh, M. H.; Jolany, H. (2013), "The Hyper-Wiener Index of One-pentagonal Carbon Nanocone", Current Nanoscience, 9 (4): 557–560, arXiv:1212.4411, doi:10.2174/15734137113090990061, S2CID 42052955.