Chemistry Reference
In-Depth Information
3. The eccentric connectivity index of G[p, q] is given by
q
⎧
⎨
+
1
+
+
+
≤
12p(q
1)(p
q
1)
p
2
ξ
(G[p, q])
=
p
>
q
.
1)
q
⎩
4p
q
2
+
+
1
−
(
−
+
1
4
2
6.5
Conclusions
We have presented here a review of topological-based methods for evaluating the
symmetry constraints and topological indices for carbon nanostructures such as nan-
otubes and nanotori. The first task, has been fulfilled by considering the
topological
symmetry
of the graph of a given chemical system; such a symmetry just reflects the
symmetry properties of the automorphism group of the graph providing
the upper
limit—fully rooted in topology—on the geometrical symmetry
of the nanostructure.
The second goal considers the application of topological indices in characterizing
nanostructures. The reported theorems are very useful because they allow a fast com-
putation of the topological indices for complex graphs (nanostructures) starting from
their structural building elements, to derive exact algorithms easy to programme on
the computer.
While the primary application concerns the description of carbon-networks in
chemical compounds, other applications exist to rank for example proteins according
to their degree of folding and topological invariants are useful in social-networks
description.
Acknowledgements
The first and second authors are partially supported by the university of
Kashan under grant number 364988/99.
References
Arezoomand M, Taeri B (2009) The full symmetry and irreducible representations of nanotori. Acta
Cryst A 65:249-252
Asadpour J, Mojarad R, Safikhani L (2011) Computing some topological indices of nanostructures.
Dig J Nanomat Biostruct 6:937-941
Ashrafi AR, Loghman A (2006a) PI index of armchair polyhex nanotubes. Ars Combin 80:193-199
Ashrafi AR, Loghman A (2006b) PI index of zig-zag polyhex nanotubes. MATCH Commun Math
Comput Chem 55:447-452
Ashrafi AR, Loghman A (2006c) Padmakar-Ivan index of TUC
4
C
8
(S) nanotubes. J Comput Theor
Nanosci 3:378-381
Ashrafi AR, Loghman A (2008) Computing Padmakar-Ivan index of a TC
4
C
8
(R) Nanotorus. J
Comput Theor Nanosci 5:1431-1434
Ashrafi AR, Rezaei F (2007) PI index of polyhex nanotori. MATCH Commun Math Comput Chem
57:243-250
