Steiner distance based and wiener type indices for graphs

Abstract

Chemical graph theory is concerned with all aspects of the application of graph theory to Chemistry. A chemical graph is a model of a chemical system, used to characterize the interactions among its components: atoms, bonds, groups of atoms or molecules. A structural formula of a molecular compound can be represented by a molecular graph, its vertices being atoms and edges corresponding to covalent bonds. Correlating and predicting physical, chemical and biological activity/property from molecular structure is very important and are unsolved problems in theoretical and computational chemistry, environmental chemistry, medical chemistry and life science. A Topological index is a numerical parameter of the molecular graph that is directly correlated with the Quantitative Structure Activity Relationships (QSAR) and Quantitative Structure Property Relationships (QSPR) of the molecule. Most of the topological indices involve graph distance or degree of the vertex. In general many of the topological indices are algebraic sum or multiplicative functions with respect to vertices and edges of a molecular graph. The Wiener index is the distance based topological index used in chemistry which was introduced by Harold Wiener in 1947. Wiener index is defined as the sum of the distance between any two carbon atoms in the molecule. The Steiner distance in a graph is a natural generalization of the concept of classical graph distance. For a connected graph G of order at least two and S c V (G) the Steiner distance dG (S) of the vertices of S is the minimum size of a connected subgraph whose vertex set is S. newline