Iranian Journal of Mathematical Chemistry
Year:2014 | Volume:5 | Issue:1 (SUPPLEMENT)|Papers Count:7

Dendrimers are highly branched organic macromolecules with successive layers or generations of branch units surrounding a central core [1, 4]. These are key molecules in nanotechnology and can be put to good use. In this article, we compute the first geometric-arithmetic index of two infinite classes of dendrimers.

The k-th semi total point graph of a graph G, Rk (G), is a graph obtained from G by adding k vertices corresponding to each edge and connecting them to the endpoints of edge considered. In this paper, a formula for Laplacian polynomial of Rk (G) in terms of characteristic and Laplacian polynomials of G is computed, where G is a connected regular graph. The Kirchhoff index of Rk (G) is also computed.

Suppose G is an n-vertex and m-edge simple graph with edge set E(G). An integer-valued function f: E(G)→Z is called a flow. Tutte was introduced the flow polynomial F(G, l) as a polynomial in an indeterminate l with integer coefficients by F(G, l) = (−1)|E(G)| S SÌE(G)(-1)|S|ln-m+c(G:S), where c(G:S) is the number of connected components of G and (G:S) denotes the spanning subgraph of G with edge set S. In this paper the Flow polynomial of some dendrimers are computed.

The neighbourhood polynomial N(G, x) , is generating function for the number of faces of each cardinality in the neighbourhood complex of a graph. In other word N(G, x)=SUÎN(G)X|U|, where N(G) is neighbourhood complex of a graph, whose vertices are the vertices of the graph and faces are subsets of vertices that have a common neighbour. In this paper we compute this polynomial for some nanostructures.

We find recursive formulae for the number of perfect matchings in a graph G by splitting G into subgraphs H and Q. We use these formulas to count perfect matching of P hypercube Qn. We also apply our formulas to prove that the number of perfect matching in an edge-transitive graph is Pm(G)=(2q/p)Pm(G\{u, n})where Pm(G) denotes the number of perfect matchings in G, G\{ u, n} is the graph constructed from G by deleting edges with an end vertex in {u, n} and unÎE(G).

The center (periphery) of a graph is the set of vertices with minimum (maximum) eccentricity. In this paper, the structure of centers and peripheries of some classes of composite graphs are determined. The relations between eccentricity, radius and diameter of such composite graphs are also investigated. As an application we determine the center and periphery of some chemical graphs such as nanotorus and nanotubes covered by C4.

In theoretical chemistry, molecular structure descriptors are used to compute properties of chemical compounds. Among them Wiener, Szeged and detour indices play significant roles in anticipating chemical phenomena. In the present paper, we study these topological indices with respect to their difference number.