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

Area of Science:

• Graph Theory
• Abstract Algebra
• Chemical Graph Theory

Background:

• Topological indices quantify molecular structures, predicting physicochemical and thermodynamic properties.
• Non-commuting graphs represent finite group structures, with vertices as non-central elements and edges indicating non-commutation.

Purpose of the Study:

• To investigate topological properties of non-commuting graphs of finite groups.
• To analyze Hosoya characteristics of non-commuting graphs over finite subgroups of SL(2,C).
• To calculate the Hosoya index for non-commuting graphs of binary dihedral groups.

Main Methods:

• Calculation of various topological indices including Harary, harmonic, Randić, reciprocal Wiener, atomic-bond connectivity, and geometric-arithmetic indices.
• Analysis of Hosoya polynomial and reciprocal status Hosoya polynomial for non-commuting graphs.
• Specific computation of the Hosoya index for non-commuting graphs derived from binary dihedral groups.

Main Results:

• Several topological indices were investigated for non-commuting graphs.
• Hosoya characteristics were analyzed for specific non-commuting graph structures.
• The Hosoya index was successfully computed for non-commuting graphs of binary dihedral groups.

Conclusions:

• The study successfully applies topological indices to non-commuting graphs, extending graph-theoretic analysis.
• The findings contribute to understanding the structural properties of finite groups through graph theory.
• This research provides a foundation for further exploration of topological indices in abstract algebra and chemical graph theory.