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This paper aims to present Bipolar valued probabilistic hesitant fuzzy sets (BVPHFSs) by combining bipolar fuzzy sets and probabilistic hesitant fuzzy sets (PHFSs). PHFSs are a strong version of hesitant fuzzy sets (HFSs) in terms of evaluated as probabilistic of each element. Probabilistic hesitant fuzzy sets (PHFSs) are a set structure that argues that each alternative should be evaluated probabilistically. In this framework, the proposed cluster allows probabilistic evaluation of decision- makers' opinions as negative. Thus, this case proposes flexibility about selection of an element and aids to overcome with noise channels. Furthermore, some new aggregation operators are discussed called bipolar valued probabilistic hesitant fuzzy weighted average operator (BVPHFWA), Generalized bipolar valued probabilistic hesitant fuzzy weighted average operator (GBVPHFWA), bipolar valued probabilistic hesitant fuzzy weighted geometric operator (BVPHFWG), Generalized bipolar valued probabilistic hesitant fuzzy weighted geometric operator (GBVPHFWG), bipolar valued probabilistic hesitant fuzzy hybrid weighted arithmetic and geometric operator (BVPHFHWAG) and Generalized bipolar valued probabilistic hesitant fuzzy hybrid weighted arithmetic and geometric (GBVPHFHWAG) and some basic properties are presented. A score function is defined ranking alternatives. Moreover, two different algorithms are put forward with helping to TOPSIS method and by using aggregation operators over BVPHFSs. The validity of proposed operators are analyzed with an example and results are compared in their own. [ABSTRACT FROM AUTHOR]

Hypersoft sets (HSSs) have gained popularity because of their ability to formulate data in the form of several trait-valued disjoint sets that blend various traits. Motivated by this idea, in this study, we present a new hyper-approach referred to as bipolar neutrosophic hypersoft sets (BNHSSs) by a generalization of neutro-sophic hypersoft sets (NHSSs) and bipolar fuzzy hypersoft sets (BFHSSs) or by merging and subjecting both HSSs and neutrosophic sets (NSs) to a bipolarity property of real numbers. By utilizing positive and negative neutrosophic structures, we construct different notions and operations on the basis of BNHSSs, such as an absolute BNHSS, a null BNHSS, a complement, subset-hood, a restricted and extended union, and a restricted and extended intersection, along with their related properties. Also, some operations like OR and AND on BNHSS have been initiated. In addition, some properties are displayed paired together, and some numerical hypothetical examples are given to clarify the mechanism of using these instruments. Finally, to prove the efficiency and applicability of the proposed model, we established two novel algorithms based on mathematical techniques (aggregation operator and score function) applied to our model (BNHSS). The aforementioned meth-ods have been utilized in the resolution of a multi-attribute group decision-making (MAGDM) problem. Some discussions and comparisons between the given techniques are also presented to demonstrate their effectiveness and applicability. [ABSTRACT FROM AUTHOR]

We establish the hybrid concept of complex bipolar- valued neutrosophic soft set (CBVNSS) as a hybrid model of bipolar neutrosophic soft set (BNSS)and complex fuzzy set (CFS). A CBVNSS is highly suitable for use in real life situations where the decision makers are interested to deal with bipolarity as well as truth membership, indeterminacy membership and falsity membership grades to the alternatives in an extended range with complex numbers. Certain operations on CBVNSS like complement, subset, union and intersection operations are defined. Some related examples are also given to enhance the understanding of the proposed concept. The basic properties are also verified. We then provide a decision-making method on the CBVNSS. Finally, a numerical example has been presented to verify validity and feasibility of the developed method. [ABSTRACT FROM AUTHOR]

In this paper, the notion of the interval valued complex neutrosophic soft set (IV-CNSS) is defined as a combination of interval valued complex neutrosophic set and soft set. Then, we introduce the IV-CNSS's operations such as union, intersection and complement. To explore the study further, we present some basic operational rules, and investigate their properties. A new algorithm is developed by transforming the interval complex neutrosophic soft values from complex space to the real space using a practical formula which gives a decision-making with a simple computational process without the need to carry out directed operations by complex numbers. This algorithm is then applied to evaluate two kinds of a certain product from a manufacturer and choose the most suitable one. IV-CNSS provides an interval-based membership structure to handle the uncertain data. This feature allows decision makers to record their hesitancy in assigning membership values which in turn best catch the obscurity and the complexity of such data. [ABSTRACT FROM AUTHOR]

Renewable energy sources play an influential role in the world's climate and reduce the rate of harmful gasses such as carbon dioxide, methane, nitrous oxide, and many other greenhouse gasses that contribute to global warming. The theoretical concept of the T-spherical fuzzy (T-SF) set (T-SFS) is the most suitable model to evaluate energy resources under uncertainty. This article illustrates appropriate operations based on Dombi triangular norm and t-conorm. We derived a series of new aggregation approaches, such as T-SF Dombi Hamy mean (T-SFDHM) and T-SF weighted Dombi Hamy Mean (T-SFDWHM) operators. Further authors illustrated a list of new approaches such as T-SF Dual Dombi Hamy mean (T-SFDDHM), and T-SF Dombi weighted Dual Hamy mean (T-SFDWDHM) operators. Some exceptional cases and desirable properties of our derived approaches are also studied. We illustrate an application of renewable energy resources to be evaluated using a multi-attribute group decision-making (MAGDM) method. A case study was also studied to choose appropriate energy resources using our proposed methodology of the T-SFDWHM and T-SFDWDHM operators. To show the effectiveness and validity of our current methods, we compared the existing results with currently developed aggregation operators (AOs). [ABSTRACT FROM AUTHOR]

This research work proposes a novel approach for multi stage decision analysis (MSDA) using innovative concepts of cubic intuitionistic fuzzy set (CIFS) theory. The paper introduces CIF-technique for order preference by similarity to ideal solution (TOPSIS) as a robust method for MSDA problems, particularly for the diagnosis of epilepsy disorders. To achieve this goal, new similarity measures (SMs) are developed for CIFS, including the Cosine angle between two vectors, a new distance measure, and the Cosine function, presented as three different types of Cosine similarity measures. The proposed CIF-TOPSIS approach is found to be suitable for precise value performance ratings and is expected to be a viable approach for case studies in the diagnosis of epilepsy disorders. The efficiency and reliability of the proposed MSDA methods is efficiently carried through numerical examples and comparative analysis. [ABSTRACT FROM AUTHOR]

The notion of classical soft multisets is extended to neutrosophic vague soft multisets by applying the theory of soft multisets to neutrosophic vague sets to be more effective and useful. We also define its basic operations, namely complement, subset, union, intersection along with illustrative examples, and study some related properties with supporting proofs. Lastly, this notion is applied to a decision making problem and its effectiveness is demonstrated using an illustrative example. [ABSTRACT FROM AUTHOR]

This paper introduces a novel soft computing technique, called the complex neutrosophic soft expert relation (CNSER), to evaluate the degree of interaction between two hybrid models called complex neutrosophic soft expert sets (CNSESs). CNSESs are used to represent two-dimensional data that are imprecise, uncertain, incomplete and indeterminate. Moreover, it has a mechanism to incorporate the parameter set and the opinions of all experts in one model, thus making it highly suitable for use in decision-making problems where the time factor plays a key role in determining the final decision. The complex neutrosophic soft expert set and complex neutrosophic soft expert relation are both defined. Utilizing the properties of CNSER introduced, an empirical study is conducted on the relationship between the variability of the currency exchange rate and Malaysian exports and the time frame (phase) of the interaction between these two variables. This study is supported further by an algorithm to determine the type and the degree of this relationship. A comparison between different existing relations and CNSER to show the ascendancy of our proposed CNSER is provided. Then, the notion of the inverse, complement and composition of CNSERs along with some related theorems and properties are introduced. Finally, we define the symmetry, transitivity and reflexivity of CNSERs, as well as the equivalence relation and equivalence classes on CNSESs. Some interesting properties are also obtained. [ABSTRACT FROM AUTHOR]

This paper presents a novel complex neutrosophic soft expert set (CNSES) concept. The range of values of CNSES is extended to the unit circle in the complex plane by adding an additional term called the phase term which describes CNSES's elements in terms of the time aspect. CNSES is a hybrid structure of soft sets and single-valued neutrosophic sets (SVNSs) defined in a complex setting where the experts' opinions are included, thus making it highly suitable for use in decision-making problems that involve uncertain and indeterminate data where the time factor plays a key role in determining the final decision. Based on this new concept we define some concepts related to this notion as well as some basic operations namely the complement, union, intersection, AND and OR. The basic properties and relevant laws pertaining to this concept such as the De Morgan's laws are also verified. Lastly, we propose an algorithm to solve complex neutrosophic soft expert decision-making problem by converting it from the complex state to the real state and subsequently provided the detailed decision steps. This study is supported by the comparison with other existing methods. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce the concept of neutrosophic vague soft expert sets (NVSESs for short) which combines neutrosophic vague sets and soft expert sets to be more effective and useful. We also define its basic operations, namely complement, union, intersection, AND and OR along with illustrative examples, and study some related properties with supporting proofs. Lastly, this concept is applied to a decision making problem and its effectiveness is demonstrated using a hypothetical example. [ABSTRACT FROM AUTHOR]

A Q-neutrosophic soft environment is an innovative hybrid tool that combines features of both a Q-neutrosophic set (Q-NS) and a parametric tool "soft set" (SS) in order to manage imprecise and indeterminate situations in various mathematical problems. In this article, we introduce a new algebraic approach called Q-neutrosophic soft matrices (Q-NSMs) to address the issues of two-dimensional (two variables) in a universal set by representing the concept of Q-neutrosophic soft sets (Q-NSSs) in matrices. On Q-NSMs, we define the fundamental set operations and some algebraic operations, i.e., complement, union, intersection, addition, subtraction, multiplication, and scalar multiplication, and prove related properties of these operations. Moreover, these operations are illustrated via several numerical examples. Then, two algorithms are proposed to tackle group decision making (GDM) problems: The first depends on the score function of Q-NSMs, and the second is based on the aggregation operator of Q-NSMs. Finally, this study is supported by a brief comparison with some relevant previous models. [ABSTRACT FROM AUTHOR]

Interval complex neutrosophic soft set (ICNSS) is the generalization of complex neutrosophic soft set (CNSS) as it provides an interval-based membership structure to handle the complex neutrosophic soft data. However, in the definition of the ICNSS, parameters set is a classical set, and the parameters have the same degree of importance which is considered as 1. This poses a limitation in modeling of some problems. Therefore, we introduce the concept of fuzzy parameterized interval complex neutrosophic soft set (FP-ICNSS) based on idea that each of elements of parameters set has got an importance degree. The basic theoretical operations and properties are defined and verified on FP-ICNSS. For FP-ICNSS, we conceptualize the relevant mapping and study the properties of the FP-ICNSS images and inverse images. Then, we propose a new algorithm that is applicable in the field of medical diagnosis and decision-making problems for selection right product. Moreover, an illustrative example is presented which depicts its validity for successful application to the problems involving vagueness and uncertainties. Eventually, a comparison between the proposed model and the existing methods is conducted to clarify the importance of this model. [ABSTRACT FROM AUTHOR]

Inequalities serve as fundamental tools for analyzing various important concepts in stochastic differential problems. In this study, we present results on the existence, uniqueness, and averaging principle for fractional neutral stochastic differential equations. We utilize Jensen, Burkholder–Davis–Gundy, Grönwall–Bellman, Hölder, and Chebyshev–Markov inequalities. We generalize results in two ways: first, by extending the existing result for p = 2 to results in the L p space; second, by incorporating the Caputo–Katugampola fractional derivatives, we extend the results established with Caputo fractional derivatives. Additionally, we provide examples to enhance the understanding of the theoretical results we establish. [ABSTRACT FROM AUTHOR]

The q-rung orthopair fuzzy set (q-ROFS) is a moderate mathematical model, that has diverse capabilities to handle uncertain and ambiguous information of human opinion during the decision analysis process. The Aczel Alsina operations are more flexible and valuable aggregating tools with parameter values ϻ ⩾ 1, reflecting smooth and accurate information by aggregating awkward and redundant information. The theory of the Choquet integral operator is also used to express the interaction between preferences or criteria by incorporating certain values of preferences. The primary features of this article are to derive some dominant mathematical approaches by combining two different theories like Choquet integral operators and operations of Aczel Alsina tools namely “q-rung orthopair fuzzy Choquet integral Aczel Alsina average” (q-ROFCIAAA), and “q-rung orthopair fuzzy Choquet integral Aczel Alsina geometric” (q-ROFCIAAG) operators. Some special cases and notable characteristics are also demonstrated to show the feasibility of derived approaches. Based on our derived aggregation approaches, a multi-attribute decision-making (MADM) technique aggregates redundant and unpredictable information. In light of developed approaches, a numerical example study to evaluate suitable safety equipment in the construction sector. To reveal the intensity and applicability of derived approaches by contrasting the results of prevailing approaches with currently developed AOs. [ABSTRACT FROM AUTHOR]

In the study of biological systems, nonlinear models are commonly employed, although exact solutions are often unattainable. Therefore, it is imperative to develop techniques that offer approximate solutions. This study utilizes the Elzaki residual power series method (ERPSM) to analyze the fractional nonlinear smoking model concerning the Caputo derivative. The outcomes of the proposed technique exhibit good agreement with the Laplace decomposition method, demonstrating that our technique is an excellent alternative to various series solution methods. Our approach utilizes the simple limit principle at zero, making it the easiest way to extract series solutions, while variational iteration, Adomian decomposition, and homotopy perturbation methods require integration. Moreover, our technique is also superior to the residual method by eliminating the need for derivatives, as fractional integration and differentiation are particularly challenging in fractional contexts. Significantly, our technique is simpler than other series solution techniques by not relying on Adomian's and He's polynomials, thereby offering a more efficient way of solving nonlinear problems. [ABSTRACT FROM AUTHOR]

Interval complex neutrosophic soft sets (I-CNSSs) refers to interval neutrosophic soft sets (I-NSSs) featuring three two-dimensional independent membership functions accordingly (falsity, indeterminacy, as well as uncertainty interval). A relation is a tool that helps in describing consistency and agreement between objects. Throughout this paper, we insert and discuss the interval complex neutrosophic soft relation (simply denoted by I-CNSR), a novel soft computing technique used to examine the interaction degree among corresponding models known as I-CNSSs. We present the definition of the Cartesian product of I-CNSSs followed by the definition of I-CNSR. Furthermore, the definitions and some theorems and properties related to the composition, inverse, and complement of I-CNSR are provided. The notions of symmetric, reflexive, transitive, and equivalent of I-CNSRs are proposed, and the algebraic properties of these concepts are verified. Furthermore, we demonstrate the relevance of our notion to real-world situations by offering a suggested method for solving a decision-making issue in the field of economics. Ultimately, an analysis is made between the current relationships and the proposed model to determine the model's significance. [ABSTRACT FROM AUTHOR]

The idea of an interval complex neutrosophic soft set (I-CNSS) emerges from the interval neutrosophic soft set (I-NSS) by the extension of its three membership functions (T, I, F) from real space to complex space (unit disc) to better handle uncertainties, vagueness, indeterminacy, and imprecision of information in the periodic nature. The novelty of I-CNSS lies in its more significant range of activity compared to CNS. Measures of similarity and distance are important tools that can be used to solve many problems in real life. Hence, this paper presents some interval complex neutrosophic soft similarities based on Hamming and Euclidean distances of I-CNSSs to deal with real-life problems that include uncertain information such as decision-making issues and medical diagnosis stats. Firstly, this paper reviews the definition of an interval complex neutrosophic soft set. Secondly, we defined distance Hamming measures and distance Euclidean measures on I-CNSSs. Next, the axiomatic definition of similarity measures based on Hamming and Euclidean distances of I-CNSSs is proposed. Moreover, a numerical example is given and relations between these measures are introduced and verified. Meanwhile, some applications are given to show how similarity can be used to help the user in making decisions and making medical diagnoses. Finally, a comparison of some current approaches is used to back up this study. [ABSTRACT FROM AUTHOR]

Nowadays, real world problems are complicated because they deal with uncertainty and incomplete information. Obviously, such problems cannot be solved by a single technique because of the multiple perspectives that may arise. Currently, the combination of DEMATEL and the neutrosophic environment are still new and not fully explored. Previous studies of DEMATEL and this neutrosophic environment have been carried out based on numerical values to represent a new scale. Until now, little importance has been placed on the development of a linguistic variable for DEMATEL. It is important to develop a new linguistic variable to represent opinions based on human experience. Therefore, to fill this gap, the concept of Interval Neutrosophic Vague Sets (INVS) has been extended to the linguistic variable that can be used in the decision-making process. The INVS is useful tool to deal with uncertainty and incomplete information. Additionally, the advantages of the linguistic variable of INVS allows the greater range of value for membership functions. This study proposes a new framework for INVS and DEMATEL. In addition, a case study on the quality of hospital service has been evaluated to demonstrate the proposed approach. Finally, a comparative analysis to check the feasibility of the proposed method is presented. It demonstrates that different methods produce different relations and levels of importance. This is due to the inclusion of the INVS linguistic variable. [ABSTRACT FROM AUTHOR]

Supplier selection is a very crucial process within a business or commercial enterprise because it depends upon different components like reliability, customer need, services, cost and reputation. A suitable supplier is familiar with developing a relationship between customer needs and business. To serve this purpose, the multiple attribute group decision-making (MAGDM) technique is a well-known and efficient aggregation model used to evaluate flexible optimal options by considering some appropriate criteria or attributes. Experts face some sophisticated challenges during the decision-making process due to uncertain and ambiguous information about human opinions. To address such conditions, we explore the notion of spherical fuzzy sets (SFS) and their reliable operations. Some flexible operational laws of Dombi t-norms are also developed in light of spherical fuzzy (SF) information. Combining the theory of Hamy mean (HM) models and Dombi aggregation tools, some robust strategies are also studied in this research work. The main objectives of this article are to propose some dominant strategies in the presence of SF information including spherical fuzzy Dombi Hamy mean (SFDHM), spherical fuzzy Dombi weighted Hamy mean (SFDWHM), spherical fuzzy Dombi Dual Hamy mean (SFDDHM) and spherical fuzzy Dombi weighted Dual Hamy mean (SFDWDHM) operators. The MAGDM techniques are utilized to evaluate the flexibility of our derived methodologies under considering SF information. An experimental case study is utilized to evaluate a notable supplier enterprise under consideration of our developed methodologies. Finally, a comprehensive overview of our research work is also presented. [ABSTRACT FROM AUTHOR]

This paper focuses on the power graph of dihedral groups of order 2n, D2n, where n ≥ 3. We show the characteristic polynomial of the power graph corresponding to the adjacency, Laplacian, signless Laplacian, and normalized form of these matrices. [ABSTRACT FROM AUTHOR]

In this research work, we use the concepts of contraction mapping to establish the existence and uniqueness results and also study the averaging principle in L p space by using Jensen's, Grönwall–Bellman's, Hölder's, and Burkholder–Davis–Gundy's inequalities, and the interval translation technique for a class of fractional neutral stochastic differential equations. We establish the results within the framework of the Ψ -Caputo derivative. We generalize the two situations of p = 2 and the Caputo derivative with the findings that we obtain. To help with the understanding of the theoretical results, we provide two applied examples at the end. [ABSTRACT FROM AUTHOR]

Introduction: The soft set theory has drawn the attention of many researchers, particularly for dealing with uncertainty in decision-making problems. Despite its remarkable advantages, the soft set theory has only been used to tackle decision-making problems that aim to choose the best option. However, there exist different forms of decision-making problems that involve different forms of uncertainty. Methods: In this study, we present various algorithms based on the soft set theory in order to handle the cases where one has different uncertainty forms in decision-making problems. Some new concepts such as object code, personal object code, parameter significance weight and new distance measures have been introduced to the literature for the construction of these algorithms. Furthermore, we show the application results of those algorithms and provide several examples. Results and Conclusions: As a result, a comparison among the application results of the algorithms implies that the best objects might not always yield the most efficient outcomes. [ABSTRACT FROM AUTHOR]

Multiple-attribute group decision-making (MAGDM) technique is often used to make decisions when several optimal options are under consideration. It can be difficult to select a reasonable optimal option for the decision maker under consideration of insufficient information. The theory of Hamy mean (HM) operators are used to express correlation among different input arguments and provide a smooth approximation during the decision-making process. Recently, Aczel Alsina aggregating expressions gained a lot of attention from numerous mathematicians under different fuzzy circumstances. This article aims to illustrate the notion of a Pythagorean fuzzy (PyF) set (PyFS) with some restricted constraints, such as a sum of the square of truth membership value and falsity membership value. We developed a series of new approaches under consideration of the HM tools, including PyF Aczel Alsina Hamy mean (PyFAAHM), and PyF Aczel Alsina weighted Hamy mean (PyFAAWHM) operators. Further, we also extend the theory of Dual Hamy mean (DHM) operators and derived a series of new methodologies such as PyF Aczel Alsina Dual Hamy mean (PyFAADHM) and PyF Aczel Alsina weighted Dual Hamy mean (PyFAAWDHM) operators. To demonstrate the flexibility of our derived approaches, we illustrate an application of a multinational company considering the MAGDM technique. An experimental case study is also illustrated to evaluate a reasonable option from a group of options. We see the advantages and compatibility of our findings by comparing the results of existing approaches with the results of currently discussed methodologies. [ABSTRACT FROM AUTHOR]

Entropy measure is an important tool in measuring uncertain information and plays a vital role in solving Multi Criteria Decision Making (MCDM). At present, various entropy measures in literature are developed to measure the degree of fuzziness. However, they could not be used to deal with interval neutrosophic vague set (INVS) environment. In this study, two kinds of entropy measures are proposed as the extension of the entropy measure of single valued neutrosophic set (SVNS). First, we construct the axiomatic definition of INVS and propose a new formula for the entropy measure of INVS. Based on this measure, we develop two multi criteria decision making methods. Subsequently, an illustrative example of investment problems under INVS is given to demonstrate the proposed entropy measures. Finally, a comparative analysis is presented to illustrate the rationality and effectiveness of the proposed entropy measures. [ABSTRACT FROM AUTHOR]

To handle indeterminate and incomplete data, neutrosophic logic/set/probability were established. The neutrosophic truth, falsehood and indeterminacy components exhibit symmetry as the truth and the falsehood look the same and behave in a symmetrical way with respect to the indeterminacy component which serves as a line of the symmetry. Soft set is a generic mathematical tool for dealing with uncertainty. Rough set is a new mathematical tool for dealing with vague, imprecise, inconsistent and uncertain knowledge in information systems. This paper introduces a new rough set model based on neutrosophic soft set to exploit simultaneously the advantages of rough sets and neutrosophic soft sets in order to handle all types of uncertainty in data. The idea of neutrosophic right neighborhood is utilised to define the concepts of neutrosophic soft rough (NSR) lower and upper approximations. Properties of suggested approximations are proposed and subsequently proven. Some of the NSR set concepts such as NSR-definability, NSR-relations and NSR-membership functions are suggested and illustrated with examples. Further, we demonstrate the feasibility of the newly rough set model with decision making problems involving neutrosophic soft set. Finally, a discussion on the features and limitations of the proposed model is provided. [ABSTRACT FROM AUTHOR]

In the definition of the complex neutrosophic soft expert set (CNSES), parameters set is a classical set, and the parameters have the same degree of importance, which is considered as 1. This poses a limitation in modeling of some problems. This paper introduces the concept of fuzzy parameterized complex neutrosophic soft expert set (FP-CNSES) to handle this issue by assigning a degree of importance to each of the problem parameters. We further develop FP-CNSES by establishing the concept of weighted fuzzy parameterized complex neutrosophic soft expert set (WFP-CNSES) based on the idea that each expert has a relative weight. These new mathematical frameworks reduce the chance of unfairness in the decision making process. Some essential operations with their properties and relevant laws related to the notion of FP-CNSES are defined and verified. The notation of mapping on fuzzy parameterized complex neutrosophic soft expert classes is defined and some properties of fuzzy parameterized complex neutrosophic soft expert images and inverse images was investigated. FP-CNSES is used to put forth an algorithm on decision-making by converting it from complex state to real state and subsequently provided the detailed decision steps. Then, we provide the comparison of FP-CNSES to the current methods to show the ascendancy of our proposed method. [ABSTRACT FROM AUTHOR]

The basic aim of soft computing is to trade precision for a tractableness and reduction in solution cost by pushing the limits of tolerance for imprecision and uncertainty. This paper introduces a novel soft computing technique called complex neutrosophic relation (CNR) to evaluate the degree of interaction between two complex neutrosophic sets (CNSs). CNSs are used to represent two-dimensional information that are imprecise, uncertain, incomplete and indeterminate. The Cartesian product of CNSs and subsequently the complex neutrosophic relation is formally defined. This relation is generalised from a conventional single valued neutrosophic relation (SVNR), based on CNSs, where the ranges of values of CNR are extended to the unit circle in complex plane for its membership functions instead of [0, 1] as in the conventional SVNR. A new algorithm is created using a comparison matrix of the SVNR after mapping the complex membership functions from complex space to the real space. This algorithm is then applied to scrutinise the impact of some teaching strategies on the student performance and the time frame(phase) of the interaction between these two variables. The notion of inverse, complement and composition of CNRs along with some related theorems and properties are introduced. The performance and utility of the composition concept in real-life situations is also demonstrated. Then, we define the concepts of projection and cylindric extension for CNRs along with illustrative examples. Some interesting properties are also obtained. Finally, a comparison between different existing relations and CNR to show the ascendancy of our proposed CNR is provided. [ABSTRACT FROM AUTHOR]

This paper focuses on the non-commuting graph for dihedral groups of order 2n, D2n, where n ≥ 3. We show the spectrum and energy of the graph corresponding to the closeness matrix. The result is that the obtained energy is always twice its spectral radius and is never an odd integer. Moreover, it is classified as hypoenergetic. [ABSTRACT FROM AUTHOR]

Recently, vague sets and neutrosophic sets have received great attention among the scholars and have been applied in many applications. But, the actual theoretical impacts of the combination of these two sets in dealing uncertainties are still not fully explored until now. In this paper, a new generalized mathematical model called interval neutrosophic vague sets is proposed, which is a combination of vague sets and interval neutrosophic sets and a generalization of interval neutrosophic vague sets. Some definitions of interval neutrosophic vague set such as union, complement and intersection are presented. Furthermore, the basic operations, the derivation of its properties and related example are included. [ABSTRACT FROM AUTHOR]