This research study focuses on the theoretical examination of the fuzzy graph of aromatic hydrocarbon naphthalene. It introduces new definitions for the first Zagreb fuzzy index, forgotten index, and Y-index, and computes several fuzzy-based degree topological indices. The study provides a general formula for the fuzzy topology indices of naphthalene hydrocarbon by taking into account the degree of each edge. Additionally, a formula is presented to determine the specific topology index of naphthalene based on the number of rows and columns. The topological indices of linear naphthalene were compared, and it was found that the Randic and harmonic indices exhibited the highest values.