A real-number to molecular structure mapping is a topological index. It is a graph invariant method for describing physico-chemical properties of molecular structures specific substances. In that article, We examined pentacene’s chemical composition. The research on the subsequent indices is reflected in our paper, we conducted an analysis of several indices including general randic connectivity index, first general zagreb index, general sum-connectivity index, atomic bond connectivity index, geometric-arithmetic index, fifth class of geometric-arithmetic indices, hyper-zagreb index, first and second multiple zagreb indices for a four para-lines graphs of linear [n]-pentacene and multi-pentacene.

This article fundamentally aims at the proposition of new family of estimators using auxiliary information to assist the estimation of finite population mean of the study variable. The objectives are achieved by devising dual use of supplementary information through straightforward manner. The additional information is injected in mean estimating procedure by considering squared values of auxiliary variable. The utility of the proposed scheme is substantiated by providing rigorous comparative account of the newly materialized structure with the well celebrated existing family of Grover and Kaur (2014). The contemporary advents of the new family are documented throughout the article.

Linear dynamic systems with impulsive effects are considered. For such a system we define a new impulsive exponential matrix. Necessary and sufficient conditions for exponential stability and boundedness have been established. The fundamental tool is an impulsive exponential matrix for exponential stability.

We study the problem of dichotomy and boundedness for impulsive dynamic equations on arbitrary closed subset of real numbers. The spectral decomposition theorem gives all our main results. The obtained results are fundamentally new, even for the classical case.

<abstract><p>This manuscript involves the new exact solitary wave solutions of fractional reaction-diffusion model using the exp $ \mathrm{(-\ }\varphi \left(\eta \right) \mathrm{)} $-expansion method. The spatial model of fractional form is applied in modeling super-diffusive systems in the field of engineering, biology, physics (neutron diffusion theory), ecology, finance, and chemistry. The findings of miscellaneous studies showed that presented method is efficient for exploring new exact solutions to solve the complexities arising in mathematical physics and applied sciences. The new solutions which are obtained in the form of the rational, exponential, hyperbolic and trigonometric functions have a wide range in physics and engineering fields. Several results would be obtained under various parameters which shows good agreement with the previous published results of different papers. The proposed method can be extended to solve further problems arising in the engineering fields. My main contribution is programming and comparisons.</p></abstract>