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: see text] of a finite group [Formula: see text] is a graph where non-central elements of [Formula: see text] are its vertex set, while two different elements are edge connected when they do not commute in [Formula: see text]. 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: see text]. We then calculate the Hosoya index for non-commuting graphs of binary dihedral groups.