Abstract

This study investigates the Atom Bond Connectivity (ABC) index applied to inverse and non-commuting graphs of finite groups. The research focuses on two distinct structures: inverse graphs formed from generalized quaternion groups and non-commuting graphs derived from non-abelian finite groups. By utilizing topological index Theory, particularly the ABC index introduced by Estrada, the study aims to characterize the graph structures based on vertex degrees. The ABC index is determined using known group properties and edge classifications for the inverse graphs of generalized quaternion groups. Similarly, the ABC index of non-commuting graphs is explored by analyzing group centers and the connectivity patterns among non-central elements. The results reveal that the ABC index quantifies group connectivity, enhancing the understanding of structural properties in algebraic graph representations. This analysis contributes to the broader application of graph Theory in abstract algebra and theoretical chemistry.

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