This research provides second-order approximate Noether symmetries of geodetic Lagrangian of time-conformal plane symmetric spacetime. A time-conformal factor is of the form $e^{ϵf(t)}$ which perturbs the plane symmetric static spacetime, where $ϵ$ is small a positive parameter that produces perturbation in the spacetime. By considering the perturbation up to second-order in $ϵ$ in plane symmetric spacetime, we find the second order approximate Noether symmetries for the corresponding Lagrangian. Using Noether theorem, the corresponding second order approximate conservation laws are investigated for plane symmetric gravitational waves like spacetimes. This technique tells about the energy content of the gravitational waves.

The study of forgotten index and coindex for the molecular structures of some special chemical graphs (compounds and drugs) has proven significant in medical and pharmaceutical drug design fields by making reliable statistical conclusion about biological properties of new chemical compounds and drugs. In mathematical chemistry, finding extreme graphs with respect to topological index is an active area of research. The aim of this paper is to characterize the family of unicyclic graphs with extreme (largest and smallest) F‐coindex. Moreover, the study also contains different other properties of unicyclic graphs. All these results are based upon an alternative form of F‐coindex which is established with the help of special property in the graphs.

We are going to investigate the geodesic motion of charged particles in the vicinity of Schwarzschild anti-de-Sitter (S-AdS) spacetime with topological defects that admit temporal perturbation. We used the approximate Noether symmetry equation to insert the time conformal factor in the black hole without losing its symmetry structure. This type of insertion is necessary because the black hole radiates its energy and momentum in the form of gravitational waves and Hawking radiation. Along with the temporal perturbation, the S-AdS black hole (BH) is immersed in an external magnetic field. We conduct an in-depth examination of the dynamics of charged particles near a weakly magnetized and time conformal S-AdS BH. Our analysis involves calculating the shift in the position of the innermost circular orbit (ISCO) caused by both temporal perturbation and the presence of a magnetic field. Furthermore, we explore the influence of dark energy (DE) and angular momentum on the stability of these orbits. Additionally, we determine the effective force and escape velocity for a charged particle orbiting around the perturbed magnetized S-AdS BH. The application of time-dependent perturbation theory can extend our understanding to investigate the quasinormal modes (QNMs) of BH mergers.

Dendrimers are highly branched, well-defined macromolecules characterized by exceptional structural precision and functional adaptability. These properties render dendrimers valuable in nanotechnology, drug delivery, and catalysis. However, selecting the most suitable dendrimer for drug delivery is difficult because its branching pattern, structural, and physicochemical properties are closely related. In this research, we utilized the multi-criteria decision-making (MCDM) technique to identify potential candidates for drug delivery applications among the four considered families: phosphorus-containing, PDI-cored, Triazine-based, and Aliphatic polyamide dendrimers. Both qualitative and quantitative molecular descriptors are integrated within the model. Degree-based topological indices, including the Narumi Katayama (NK), Modified Narumi Katayama (NK $$^*$$ ∗ ), Forgotten Index (F-index) and Augmented Zagreb Index (AZI) indices, are utilized to capture variations in molecular complexity, stability, and branching efficiency. The evaluation was conducted using entropy-based weights obtained from representative benzene derivatives; also, the equal weighting and sensitivity analyses were utilized to validate the ranking results. This study uses multi-objective optimization on the basis of ratio analysis (MOORA), simple additive weighting (SAW), and the technique for order preference by similarity to ideal solution (TOPSIS) techniques for criteria weighting, ranking, and reflects how topological descriptors support decision-making in dendrimer selection. The results of these investigations provide quantitative guidance for chemists and pharmaceutical researchers on the forecasting capacity of topological indices in designing dendrimers with enhanced drug delivery. Graphical Dendrimer structures, graph construction, multi-criteria decision-making, sensitivity analysis, property modeling, and interpretation for dendrimer applications.