Photonic crystals in biology

P –12
Poster Session, Tuesday, June 15
Theme A1 - B702
Computing Closed Walks of Nanostar Dendrimers
G. H. Fath-Tabar 1 * andA. R. Ashrafi 2
1 Department of Mathematics, Faculty of Science, University of Kashan, Kashan 87317-51167, I. R. Iran
2 Institute of Nanoscience and Nanotechnology, University of Kashan, Kashan, Iran
Abstract-Suppose G = (V, E) is a simple graph. The sequence of vertices v 0 v 1 …v t v 0 is called a closed walk if v i v i+1 are in E(G). In this paper,
the number of closed walks (CW(G, k)), k = 1,2,…, 10 and k = 11, 13, 15, … for three types of nanostar dendrimers are presented.
Dendrimers are highly branched macromolecules. They are
being investigated for possible uses in nanotechnology, gene
therapy, and other fields. The nanostar dendrimer is part of a
new group of macromolecules that appear to be photon
funnels just like artificial antennas. The topological study of
these macromolecules is the aim of this article.
In this paper, the word graph refers to a finite, undirected
graph without loops and multiple edges. Let G be a graph and
{v 1 , ..., v n } be the set of all vertices of G. The adjacency
matrix of G is a 01 matrix A(G) = [a ij ], where a ij is the
number of edges connecting v i and v j . The spectrum of a
graph G is the set of eigenvalues of A(G), together with their
multiplicities. Throughout this paper our notation is standard
and taken mainly from the standard book of graph theory. A
walk is a sequence of graph vertices and graph edges such that
the graph vertices and graph edges are adjacent. A closed walk
is a walk in which the first and the last vertices are the same.
A closed walk has backtracking if, in the closed walk, an edge
appears twice in immediate succession. For more information
1-3
about these concept you can see referencesP
P.
n+2
Cl ( NS1[
n],
2k
1)
0, Cl ( NS [ n],
2) 21.2 - 30,
n+2
Cl ( NS1[
n],
4) 48.2 1230.
1
*Corresponding author: ashrafi@kashanu.ac.ir
[1] G. H. Fath-Tabar, M. J. Nadjafi-Arani, M. Mogharrab and A. R.
Ashrafi, MATCH Commun. Math. Comput. 63 (2010) 145.
[2] I. Gutman, Graph Theory Notes of New York 27, 9 (1994).
[3] P.V. Khadikar and S. Karmarkar, J. Chem. Inf. Comput. Sci. 41,
934 (2001).
Figure 1. The Nanostar Dendrimer NSR2R[2].
We now consider the nanostar dendrimer NS[n], Figures 1.
Using a simple calculation, one can show that |V(NS[n])| =
n+1
n+1
18.2P and |E(NS[n])| = 21.2P
P-15. We prove that:
The number of closed walks of the nanostar dendrimer
NS[n ] are co mputed as follows:
6th Nanoscience and Nanotechnology Conference, zmir, 2010 355