Certain Topological Indices of Non-Commuting Graphs for Finite Non-Abelian Groups

A topological index is a number derived from a molecular structure (i.e., a graph) that represents the fundamental structural characteristics of a suggested molecule. Various topological indices, including the atom-bond connectivity index, the geometric–arithmetic index, and the Randić index, can be utilized to determine various characteristics, such as physicochemical activity, chemical activity, and thermodynamic properties. Meanwhile, the non-commuting graph (Formula presented.) of a finite group (Formula presented.) is a graph where non-central elements of (Formula presented.) are its vertex set, while two different elements are edge connected when they do not commute in (Formula presented.). In this article, we investigate several topological properties of non-commuting graphs of finite groups, such as the Harary index, the harmonic index, the Randić index, reciprocal Wiener index, atomic-bond connectivity index, and the geometric–arithmetic index. In addition, we analyze the Hosoya characteristics, such as the Hosoya polynomial and the reciprocal status Hosoya polynomial of the non-commuting graphs over finite subgroups of (Formula presented.). We then calculate the Hosoya index for non-commuting graphs of binary dihedral groups.