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.

Linear correlated fractional fuzzy differential equations (LCFFDEs) are one of the best tools for dealing with physical problems with uncertainty. The LCFFDEs mostly do not have unique solutions, especially if the basic fuzzy number is symmetric. The LCFFDEs of symmetric basic fuzzy numbers extend to the new system by extension and produce many solutions. The existing literature does not have any criteria to ensure the existence of unique solutions to LCFFDEs. In this study, we will explore the main causes of the extension and the unavailability of unique solutions. Next, we will discuss the existence and uniqueness conditions of LCFFDEs by using the concept of metric fixed point theory. For the useability of established results, we will also provide numerical examples and discuss their unique solutions. To show the authenticity of the solutions, we will also provide 2D and 3D plots of the solutions.

In this manuscript, the notion of a hesitant fuzzy soft fixed point is introduced. Using this notion and the concept of Suzuki-type ( $\mu ,\nu$)-weak contraction for hesitant fuzzy soft set valued-mapping, some fixed point results are established in the framework of metric spaces. Based on the presented work, some examples reflecting decision-making problems related to real life are also solved. The suggested method’s flexibility and efficacy compared to conventional techniques are demonstrated in decision-making situations involving uncertainty, such as choosing the best options in multi-criteria settings. We noted that the presented work combines and generalizes two major concepts, the idea of soft sets and hesitant fuzzy set-valued mapping from the existing literature.