Topological indices are mathematical values based on graph models of molecular structures that characterize significant properties in terms of chemical composition, reactivity, and physicochemical properties. In this paper, we are devoted to eccentricity-based indices of power graphs over finite groups and investigate their application in the context of molecular graphs. We calculated the Zagreb eccentricity indices, eccentric connectivity index, connective eccentricity index, (adjacent) eccentric distance sum index, and the Zagreb irregularity indices. In addition, we computed the Hosoya index for the mentioned graphs, which was one of the challenging aspects of this work. These findings enhance the theoretical foundation of graph-based indices and contribute to the quantitative description of molecular graphs.