chemia - Studia

STUDIA UBB. CHEMIA, LV, 4, 2010
ON ESTRADA INDEX OF TWO CLASSES OF DENDRIMERS
GHOLAM HOSSEIN FATH-TABAR a , ALI REZA ASHRAFI a ,
ANTE GRAOVAC b,c,d
ABSTRACT. Suppose G = (V, E) is a graph. The sequence v 1 v 2 …v t (v t = v 1 ) is
called a closed walk with length t − 1 in G if v i 's are in V(G) and v i v i+1 ∈ E(G).
In this paper, the number of closed walks with length k, C w (G,k), for two
classes of denderimers are computed.
Keywords: Dendrimer, closed walk. graph spectrum, Estrada index.
INTRODUCTION
Dendrimers are polymeric macromolecules composed of multiple
perfectly-branched monomers radially emanating from a central core,
Figures 1, 2. The number of branching points increases upon moving from
the dendrimer core to its surface and defines dendrimer generations. They
are being investigated for possible uses in nanotechnology, gene therapy,
and other fields [1-5].
In this paper, the word graph refers to a finite, undirected graph without
loops and multiple edges. Suppose G is a graph. The vertices and edges of G
are denoted by V(G) and E(G), respectively. A walk in G is an alternating
sequence of graph vertices and edges such that any subsequent two edges
are adjacent. A closed walk is a walk in which the first and the last vertices are
the same. We encourage to the reader to consult papers [6-11] for background
material, as well as basic computational techniques. Our notation is standard
and mainly taken from the standard book of graph theory [12].
MAIN RESULTS AND DISCUSSION
Let D 1 [n] and D 2 [n] be the molecular graphs of the dendrimers depicted
in Figures 1 and 2, respectively. In this section, some formulas are derived for
the number of closed walks of length k, C w (G,k), where 1 ≤ k ≤ 10 and G is
one of the molecular graphs D 1 [n] and D 2 [n]. For the sake of completeness,
we mention here a well-known theorem in algebraic graph theory as follows:
a Department of Mathematics, Statistics and Computer Science, Faculty of Science,University of
Kashan, Kashan 87317-51167, I. R. Iran
b Faculty of Science, University of Split, Nikole Tesle 12, HR-21000, Split, Croatia
c NMR Center, Institute »R. Boskovic«, HR-10002 Zagreb, POB 180, Croatia
d IMC, University of Dubrovnik, Branitelja Dubrovnika 29, HR-20000 Dubrovnik, Croatia