Topological indices of molecular graphs A degree based approach

Abstract

A molecular graph is a collection of points representing the atoms of the molecule and a set of lines representing the covalent bonds. These points are named vertices and the lines are named edges in graph theory language. A topological index is the graph invariant number calculated from a graph representing a molecule. Most of the proposed topological indices are related either to a vertex adjacency relationship (atom-atom connectivity) in the graph G or to topological distances in G. Topological indices play an important role in mathematical chemistry, especially Quantitative Structure Activity Relationship (QSAR) and Quantitative Structure Property Relationship (QSPR) investigations. Using molecular graphs, the chemical structure of compound may be expressed by means of various graph matrices, polynomials, spectra, special moments or topological indices. newlineThe Wiener index W(G), one of the best descriptions of molecular topology was defined by Wiener in 1947 as the sum of the shortest distances between all pairs of vertices of G. The Zagreb group indices ? of a graph G denoted by M1(G) (First Zagreb index), M2(G) (Second Zagreb index) and M3 (G) (Third Zagreb index) are defined as, newline newlineM1(G) = E degG(v) = E [degG(u) + degG(v)], newlinevEv(G) uvEE(G) newlineM2(G) = E [degG(u) x degG(v)] and newlineuvEE(G) newlineM3 (G) = E [degG(u) degG(v)]. uvEE(G) newlineThe new defined index ? is called as Lanzhou index of G as, Lz(G) = E dud u2 newlineuel7 (G) newlinewhere du denote the degree of u in G, the complement of G and du denotes the degree of u in G. newline newline