Gutman, Sombor, and Harmonic Indices of Unit Graphs in The Integer Ring Modulo with A Specific Order

The ring of integers modulo is an algebraic structure that plays an important role in various fields, such as number theory, cryptography, and number system modeling. This structure also has a strong connection to graph representation, especially in the formation of unit graphs. This research focuses on the analysis of unit graphs formed from modulo integer algebras of a certain order, which aims to formulate the general form of topological indices, namely Gutman, Sombor, Reduced Sombor, Average Sombor, and Harmonic. The research considers two cases: the ring of integer modulo and , where is an odd prime number. The results show that each index has a unique and specific mathematical pattern according to its unit graph order. These findings provide a deeper understanding of the topological and combinatorial properties of unit graphs, which may help in generalizing their topological indices.