On Topological Indices of a Graph Associated to the Direct Product of Two Groups

The topological structure or connectedness of a molecular network can be understood through the use of topological indices, which are numerical values. The topological indices are widely used in various disciplines including chemistry, mathematics, physics and many applications including the drug design and chemical property studies. In this paper, the topological indices of the non-commuting for the direct product of an abelian group and the dihedral groups are determined. The non-commuting graph is a graph where two of its vertices are adjacent if and only if they represent elements in a group that do not commute with one another. Moreover, the vertices of this graph correspond to the noncentral elements of the group. The general formula of the non-commuting graph for the direct product of an abelian group and the dihedral groups is established. Then, it is used to find its topological indices by using some preliminaries. The topological indices involve in this study are the Wiener index, Zagreb index, Szeged index and Harary index. It is found that their general formulas have relation between the order of an abelian group and the topological indices of the non-commuting graph for the dihedral groups.