Fisrt Zagreb, Gutman, and Wiener Index of The Non-Coprime Graph for Dihedral Group

Abstract

This study investigates the structure of non-coprime graphs derived from the dihedral groups of prime power order using approaches from graph theory and group theory. In molecular representation and pure mathematics, graphs serve a tool to illustrate the internal structure of systems through topological indices. A non-coprime graph is a graph formed by connecting two distinct vertices if the greatest common divisor (GCD) of their orders is not equal to one, meaning the two elements are not coprime. The primary focus of this research is to derive general formulas for the first Zagreb index, the Gutman index, and the Wiener index of non-coprime graphs constructed from the dihedral group ​. For the first Zagreb index, the cases  and  are considered, where  is a prime number and  is a positive integer. Meanwhile, for the Gutman and Wiener indices, the study focuses on the case, with  as a positive integer. This research produces explicit formulas for the first Zagreb index, Gutman index, and Wiener index of the corresponding non-coprime graphs.