The self-similar networks with complicated structures are becoming more significant in the context of in many scientific fields, and their study is considered as one of the most important and interesting kinds of research. We can thoroughly discuss the complex structures of fractal cactus \((\textrm{FC})\) and Sierpinski fractal \((\textrm{SF})\) networks by using the fractal dimension and different types of topological indices. We can also apply the topological indices based fractal dimensional analysis to a special type of molecular structure such as an armchair graphene \((\textrm{AG})\) network to analyze its self-similarity. In this study, the degree-based topological indices of 12 types are computed for three self-similar networks, to expose the structural properties of those graphs. Additionally one of the non-linear measures, the fractal dimension has been newly defined through 12 degree-based topological indices and estimated numerically using the computed structural indices to analyze the self-similarity level for the representative networks. Finally, the fractal dimension values obtained numerically using the proposed method are compared with the theoretically derived similarity dimension for three self-similar networks to validate the proposed numerical fractal dimension. These proposed measures will help reveal the structural characteristics of the chemical compounds.
