• Source: 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




W
W
(
G
)
=


1
2



∑

u
,
v
∈
V
(
G
)


(
d
(
u
,
v
)
+

d

2


(
u
,
v
)
)
,


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




WW
⁡
(

G

n


)
=
20
+


533
4


n
+


8501
24



n

2


+


5795
12



n

3


+


8575
24



n

4


+


409
3



n

5


+
21

n

6




{\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}}





References

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