کاربردهای شیمیایی نظریه گراف
Abstract
1- Introduction
2- Topological indices
3- Models
4- Conclusions
References
INTRODUCTION
Chemical graph theory applies this branch of mathematics to model molecules in order to study their various physical properties. A graph G = (V, E) consists of a set V of vertices (or nodes) and a set E of unordered pairs of distinct elements of V, which are the edges. In chemistry, the atoms of a molecule are represented by the vertices and the chemical bonds are represented by the edges. The resulting graph is often called a chemical graph. When studying alkanes which have the chemical formula CnH2n+2, the hydrogen atoms are removed from the graph resulting in what is known as a hydrogen-depleted molecular graph or a carbon tree. Since each carbon has four bonds and each hydrogen has one bond, no information about the molecule is lost by removing the hydrogen atoms. The resulting graph is in fact easier to study since the geometric structure of the alkane is more apparent. For example, a rendering of 2,2,4-trimethylpentane is given in Fig. 1. The chemical graph of 2,2,4- trimethylpentane is given in Fig. 2. The vertex set of the graph in Fig. 2 is V = {a, b, c, d, e, f , g, h}, and the edge set of this graph is given by E = {ab, bc, cd, de, bf , bg, dh}. The degree of a vertex v is the number of edges attached to vertex v and is denoted deg(v). In a molecular compound, this is the number of bonds an atom has and is defined as the valency of an atom. In Fig. 2, deg(a)= deg(f)=deg(g)=deg(h)=deg(e)=1, deg(b)=4, deg(c)=2, and deg(d)=3. Given a graph G with vertices u, v ∈ V(G), we define a u–v path as a sequence of edges from G of the form uv1, v1v2, v2v3, ... , vnv where each vertex vi ∈ V(G) for 1 ≤ i ≤ n is distinct. The length of the given u–v path is n, the number of edges in the path. The shortest path from vertex u to v is the path for which n, the number of edges in the path, is minimal. A path from vertex g to h in Fig. 2 is given by gb, bc, cd, dh and has length four. This is also the shortest path from g to h since this is the only g–h path. Given that 2,2,4-trimethylpentane is an acyclic hydrocarbon, its chemical graph contains no cycles.