Phenol-formaldehyde resin has a wide range of moldings. The phenolic resin retains properties at the freezing point; hence, it is difficult to determine its age. However, it has immense consumption in manufacturing electrical equipment due to its insulating property. There are many types of topological indices such as degree-based topological indices, distance-based topological indices, etc. Topological indices correlate some physiochemical properties of chemical compounds. In this article, the degree-based topological indices of phenol-formaldehyde resin have been determined. Furthermore, the Revan index, hyper Revan index, modified Revan index, sum connectivity Revan index, harmonic Revan index, and inverse Revan index have been calculated.
Tessellations of kekulenes and cycloarenes have a lot of potential as nanomolecular belts for trapping and transporting heavy metal ions and chloride ions because they have the best electronic properties and pore sizes. The aromaticity, superaromaticity, chirality, and novel electrical and magnetic properties of a class of cycloarenes known as kekulenes have been the subject of several experimental and theoretical studies. Through topological computations of superaromatic structures with pores, we investigate the entropies and topological characterization of different tessellations of kekulenes. Using topological indices, the biological activity of the underlying structure is linked to its physical properties in (QSPR/QSAR) research. There is a wide range of topological indices accessible, including degree-based indices, which are used in this work. With the total π -electron energy, these indices have a lot of iteration. In addition, we use graph entropies to determine the structural information of a non-Kekulean benzenoid graph. In this article, we study the crystal structure of non-Kekulean benzenoid graph K n and then calculate some entropies by using the degree-based topological indices. We also investigate the relationship between degree-based topological indices and degree-based entropies. This relationship is very helpful for chemist to study the physicochemical characterization of non-Kekulean benzenoid chemical. These numerical values correlate with structural facts and chemical reactivity, biological activities, and physical properties.
In this research article, we motivate and introduce the concept of possibility belief interval-valued N-soft sets. It has a great significance for enhancing the performance of decision-making procedures in many theories of uncertainty. The N-soft set theory is arising as an effective mathematical tool for dealing with precision and uncertainties more than the soft set theory. In this regard, we extend the concept of belief interval-valued soft set to possibility belief interval-valued N-soft set (by accumulating possibility and belief interval with N-soft set), and we also explain its practical calculations. To this objective, we defined related theoretical notions, for example, belief interval-valued N-soft set, possibility belief interval-valued N-soft set, their algebraic operations, and examined some of their fundamental properties. Furthermore, we developed two algorithms by using max-AND and min-OR operations of possibility belief interval-valued N-soft set for decision-making problems and also justify its applicability with numerical examples.
<p>We elaborate in this paper a new structure Pythagorean fuzzy<br />$N$-soft groups which is the generalization of intuitionistic fuzzy<br />soft group initiated by Karaaslan in 2013. In Pythagorean fuzzy<br />N-soft sets concepts of fuzzy sets, soft sets, N-soft sets, fuzzy<br />soft sets, intuitionistic fuzzy sets, intuitionistic fuzzy soft<br />sets, Pythagorean fuzzy sets, Pythagorean fuzzy soft sets are<br />generalized. We also talk about some elementary basic concepts and<br />operations on Pythagorean fuzzy N-soft sets with the assistance of<br />illusions. We additionally define three different sorts of<br />complements for Pythagorean fuzzy N-soft sets and examined a few<br />outcomes not hold in Pythagorean fuzzy N-soft sets complements as<br />they hold in crisp set hypothesis with the assistance of counter<br />examples. We further talked about {$(\alpha, \beta, \gamma)$-cut of<br />Pythagorean fuzzy N-soft set and their properties}. We likewise talk<br />about some essential properties of Pythagorean fuzzy N-soft groups<br />like groupoid, normal group, left and right cosets, $(\alpha, \beta,<br />\gamma)$-cut subgroups and some fundamental outcomes identified with<br />these terms. Pythagorean fuzzy N-soft sets is increasingly efficient<br />and adaptable model to manage uncertainties. The proposed models of<br />Pythagorean fuzzy N-soft groups can defeat a few disadvantages of<br />the existing statures.</p>
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