chemia - Studia

STUDIA UBB. CHEMIA, LV, 4, 2010
GEOMETRIC−ARITHMETIC INDEX: AN ALGEBRAIC APPROACH
HOSSEIN SHABANI a , ALI REZA ASHRAFI a* , IVAN GUTMAN b
ABSTRACT. The Geometric-Arithmetic (GA) index is a recently proposed
topological index in mathematical chemistry. In this paper, a group theoretical
method for computing the GA index of graphs is presented. We apply this
method to some classes of dendrimers to calculate their GA index.
Keywords: geometric-arithmetic index, dendrimer
INTRODUCTION
A molecular graph is a simple graph such that its vertices correspond
to the atoms and the edges to the bonds. Note that hydrogen atoms are
often omitted. By IUPAC terminology, a topological index is a numerical value
associated with a chemical constitution purporting for correlation of chemical
structure with various physical properties, chemical reactivity or biological
activity [1−3]. The name “topological index” was first used by Hosoya [4], in
connection with his Z index, which he used for characterizing the topological
nature of graphs.
A dendrimer is generally described as a macromolecule, which is
built up from a starting atom, such as nitrogen, to which carbon and other
elements are added by a repeating series of chemical reactions that produce
a spherical branching structure. In a divergent synthesis of a dendrimer,
one starts from the core (a multi connected atom or group of atoms) and
growths out to the periphery. In each repeated step, a number of monomers
are added to the actual structure, in a radial manner, resulting in quasi
concentric shells, called generations. In a convergent synthesis, the periphery
is first built up and next the branches (called dendrons) are connected to the
core. The stepwise growth of a dendrimer follows a mathematical progression
and its size is in the nanometer scale [5−7].
We now recall some algebraic notations that will be used throughout.
Suppose G is a graph with vertex and edge sets V(G) and E(G), respectively. If e
is an edge of G, connecting the vertices u and v then we write e = uv. For each
vertex a and b, d(a,b) denotes the length of a minimal path connecting them.
The eccentricity of a vertex x, ε(x), is defined as the maximum of {d(y,x) | y ∈ V(G)}.
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 Kragujevac, P. O. Box 60, 34000 Kragujevac, Serbia