Abstract

The Hyper-Wiener index is a widely used topological descriptor that quantifies the structural complexity of graphs, particularly those arising from algebraic structures. This paper presents a structured synthesis of key theorems related to the Hyper-Wiener index in coprime graphs, non-coprime graphs, and power graphs constructed from the integer modulo group and the dihedral group. Adopting a systematic literature review approach, we compile and restate formal results, including explicit formulas and proven properties. Each theorem is analyzed in relation to the algebraic structure of its underlying group and the resulting graph topology. Our findings highlight how group-theoretic properties—such as order, operation, and element interactions—directly impact the Hyper-Wiener index. This paper is intended to support researchers by providing a conceptual bridge between group theory and topological graph theory, and by identifying potential directions for future work.

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